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IN   MEMORIAM 
FLOR1AN  CAJORI 


'M 


■     JL 


AN  ELEMENTARY  TREATISE 


THE   INTEGRAL   CALCULUS 


By  the  same  Author. 

AN.  ELEMENTARY  TREATISE 

ON 

THE  DIFFERENTIAL  CALCULUS, 

CONTAINING 

THE  THEORY  OF  PLANE  CURVES. 


AN  ELEMENTARY  TREATISE 


THE  INTEGRAL  CALCULUS, 


CONTAINING 


APPLICATIONS  TO  PLANE  CURVES 
AND  SURFACES; 


NUMEROUS    EXAMPLES. 


BY 
BENJAMIN  WILLIAMSON,  M.A.,  F.R.S., 

FELLOW  OF  TRINITY  COLLEGE,   AND  PROFESSOR   OF  NATURAL  PHILOSOPHY 
IN  THE  UNIVERSITY  OF  DUBLIN. 


$ami\  €Mti0tt,   l&Mult  mts  ^wlar^. 

NEW  YORK: 

APPLETON    &    CO. 

1884. 


[all  rights  reserved.] 


PREFACE. 


This  book  has  been  written  as  a  companion  volume  to  my 
Treatise  on  the  Differential  Calculus,  and  in  its  construction 
I  have  endeavoured  to  carry  out  the  same  general  plan  on 
which  that  book  was  composed.  I  have,  accordingly,  studied 
simplicity  so  far  as  was  consistent  with  rigour  of  demonstra- 
tion, and  have  tried  to  make  the  subject  as  attractive  to  the 
beginner  as  the  nature  of  the  Calculus  would  permit. 

I  have,  as  far  as  possible,  confined  my  attention  to  the 
general  principles  of  Integration,  and  have  endeavoured  to 
arrange  the  successive  portions  of  the  subject  in  the  order 
best  suited  for  the  Student. 

I  have  paid  considerable  attention  to  the  geometrical  ap- 
plications of  the  Calculus,  and  have  introduced  a  number  of  the 
leading  fundamental  properties  of  the  more  important  curves 
and  surfaces,  so  far  as  they  are  connected  with  the  Integral 
Calculus.  This  has  led  me  to  give  many  remarkable  results, 
such  as  Steiner's  general  theorems  on  the  connexion  of  pedals 
and  roulettes,  Amsler's  Planimeter,  Kempe's  theorem, 
Landen's  theorems  on  the  rectification  of  the  hyperbola, 
Gtenocchi's  theorem  on  the  rectification  of  the  Cartesian  oval, 
and  others  which  have  not  been  usually  included  in  text- 
books on  the  Integral  Calculus. 

A  Chapter  has  been  devoted  to  the  discussion  of  Integrals 
of  Inertia.     For  the  methods  adopted,  and  a  great  part  of  the 


|w?,0rj1 52 


vi  Preface. 

details  in  this  Chapter,  I  am  indebted  to  the  kindness  of  Pro- 
fessor Townsend.  My  friend,  Professor  Crof  ton,  of  Woolwich, 
has  laid  me  under  very  deep  obligations  by  contributing  a 
Chapter  on  Mean  Yalue  and  Probability.  I  am  glad  to  be 
able  to  lay  this  Chapter  before  the  Student,  as  an  introduc- 
tion to  this  branch  of  the  subject  by  a  Mathematician  whose 
original  and  admirable  Papers,  in  the  Philosophical  Transac- 
tions, 1868-69,  and  elsewhere,  have  so  largely  contributed  to 
the  recent  extension  of  this  important  application  of  the 
Integral  Calculus. 

In  this  Edition  a  short  Chapter  on  Multiple  Integration 
has  been  introduced,  which  I  hope  will  be  found  a  useful 
addition  to  the  Book. 

Trinity  College, 
April,  1884. 


TABLE   OF    CONTENTS 


CHAPTER  I. 

ELEMENTAEY   FOEMS   OF   INTEGEATION. 

Page 
Integration, 1 

Elementary  Integrals, 2 

Integration  by  Substitution, 5 

..                    dx 
Integration  of   ; =, 8 

°  a  +  2bx  +  ex* 

Euler's  expressions  for  sin  6  and  cosfl  deduced  by  Integration,   .        .        .10 

-r  d% 

Integration  of    — , 14 

(x  -p)^a  +  2bx  +  ex* 

dx 

»  5J  3>  ••••:•-.  16 

(a  +  2fo  +  «s8)a 

ae       .    dd 

,,        ,,       and  - — , 17 

"         "      cos0  sm0'  ' 

dd 

"        "      a  +  bcosd 19 

Different  Methods  of  Integration, 20 

Formula  of  Integration  by  Parts, 20 

Integration  by  Rationalization, .23 

Rationalization  by  Trigonometrical  Transformations,         .         .         .         .26 

Observations  on  Fundamental  Forms, 29 

Definite  Integrals,  30 

Examples, 36 


viii  Contents. 


CHAPTER  II. 


INTEGRATION   OP   RATIONAL  FRACTIONS. 

Pack 

Rational  Fractions, 39 

Decomposition  into  Partial  Fractions, 42 

Case  of  Real  and  Unequal  Roots, 42 

Multiple  Real  Roots, 47 

Imaginary  Roots,             49 

Multiple  Imaginary  Roots, 52 

dx 

Integration  of — — , 56 

6  (x  -a)m(x  -b)'S 


dx 


58 


xm~*dx 
x"  —  i 
Examples, .  61 


CHAPTER  III. 

INTEGRATION   BY   SITCCESSIYE   REDUCTION. 

Cases  in  which  sinw0  coBnd  dd  is  immediately  integrable,  .        .        .'63 

Formulae  of  Reduction  for  J  sinm0  co&n6dd, 66 

dd 

Integration  of  tann0<?0  and -,            72 

tann0' 

Trigonometrical  Transformations, 73 

Binomial  Differentials,     .         .         .                  75 

Reduction  of  J  em*xndx, 76 

„  )x™{logx)»dx 77 

,,         „  \xn  cos  axdx,        .         .         .                 78 

,,        „  fea*cosnxdx, 79 

„       ♦,,  j  eoamx  a'mnxdx, 79 

Reduction  by  Differentiation, 80 


Contents.  ix 

Page 


Jx"*dx 
7 — ; r-i 81 
(a  +  cx)n 


(a  +  cxy 
xmdx 


f  _ 

"  J  (a  +  2bx  +  cx-)? 


f         xmdx 
"          "  J  {a  +  2bx  +  cx-)n' 
f  dx 

"  J  (a  +  b  cos  *)»' 


CHAPTER  IV. 

INTEGRATION   BY   RATIONALIZATION. 


Integration  of  —y-r  , 86 

$\x)  Va  +  2bx  +  mj2 

Examples, •        •         •  •         •         .89 


Integration  of  Monomials, ...  91 

Rationalization  of  F(x,  Va  +  2bx  +  cx'^dx, 92 

f(*)**  92 
</>(z)  Va  +  2fo  +  ca;2' 

General  Investigation, ...  97 

Case  of  Recurring  Biquadratic, 101 

Examples, .         .103 


CHAPTER  V. 

MISCELLANEOUS    EXAMPLES    OF   INTEGRATION . 

A  cos  x  +  B  sin  x  +  C  .  n 

Integral  of ; — : dx, 104: 

a  cos  x  +  b  sin  #  +  c 

Differentiation  under  the  sign  of  Integration, 107 

Integration  under  the  sign  of  Integration, 109 

„         hy  Infinite  Series, 110 

Examples, 113 


Contents. 


CHAPTER  VI. 


DEFINITE     INTEGRALS. 

Page 
Integration  regarded  as  Summation, 114 

Definite  Integrals.     Limits  of  Integration, 115 

rr  it 

Values  of  1   smnxdx  and  I    cosnxdx, 119 

Jo  Jo 


>>     » 


sinm#  cosnxdx,  when^m  and  n  are  integers,         .         .         .120 


»o 


I     e^x^dxy   when  n  is  a  positive  integer,         .        .        .        .123 


Taylor's  Theorem, 126 

Remainder  in  Taylor's  Theorem  expressed  as  a  Definite  Integral,    .        .127 

Bernoulli's  Series, 128 

Exceptional  Cases  in  Definite  Integrals, 128 

Case  where  f(x)  becomes  Infinite  at  either  Limit, 128 

When  f(x)  becomes  Infinite  between  the  Limits, 130 

Case  of  Infinite  Limits, 131 

Principal  and  General  Values  of  a  Definite  Integral,        .        .  .132 

Singular  Definite  Integrals, 134 

ffU 135 

Jo  -F(«) 

f"      x'Zm 

\     - r  dx,  where  n  >  m, 138 

J0   1  +  xM 

f"      x2m 

\     : Tdx, 139 

Jo  1  -  *2M 

Differentiation  of  Definite  Integrals, 143 

Integrals  deduced  by  Differentiation,       .         .         .  •       .         .         .         .144 

Differentiation  of  a  Definite  Integral  when  the  Limits  vary,    .         .         .  147 

Integration  under  the  Sign  J, 148 

[    e-**dx, 151 

Jo 


Contents.  xi 

Page 


.         .     153 

I   +  Z* 


gob  mxdx 

I  +x>   ' 

log(nnO)de, 154 

o 

Theorem  of  Frullani,* 155 

„  f00  tan-1  ax -tan.-1  bx  ,  „_„ 

Value  of dx,          .......  157 

Jo  x 

Eemainder  in  Lagrange's  Series, 158 

Gamma-Functions, 159 

Proof  of  Equation  B  (m,  n)  =      ,        \,  .         .        .        .         .         .161 

r(»)r(i-»)=-r^— , 162 

'  sin  Mir 

*»--''©■'© '©■■■■r(^i) 164 

Table  of  Log  T(p), 169 

Examples, 171 


CHAPTER  VII. 

AEEAS     OP    PLANE     CUKVES. 

Areas  in  Cartesian  Coordinates,      ....  ...  176 

The  Circle, 178 

The  Ellipse, 179 

The  Parabola, 180 

The  Hyperbola, 181 

Hyperbolic  Sines  and  Cosines, 182 

The  Catenary, 183 

Form  for  Area  of  a  Closed  Curve, 188 

The  Cycloid, 189 

Areas  in  Polar  Coordinates,  190 

Spiral  of  Archimedes, 194 

Elliptic  Sector — Lambert's  Theorem, 196 


*  This  theorem  was  communicated  by  Frullani  to  Plana  in  1821,  and  published  after- 
wards in  Mem.  del  Soc.  ltal.,  1828. 


xii  Contents. 

Pagk 

Area  of  a  Pedal  Curve,           ....                 ....  199 

Area  of  Pedal  of  Ellipse  for  any  Origin,           .                          .         .  201 

Steiner's  Theorem  on  Areas  of  Pedals, 201 

,,              on  Roulettes, 203 

Theorem  of  Holditch, 206 

Kempe's  Theorem, 210 

Areas  by  Approximation, 211 

Amsler's  Planimeter, 214 

Examples, 218 


CHAPTER  VIII. 

LENGTHS     OF     CTJETES. 

Rectification  in  Rectangular  Coordinates, 222 

The  Parabola— The  Catenary, 223 

The  Semi-cubical  Parabola — Evolutes, 224 

The  Ellipse, 226 

Legendre's  Theorem  on  Rectification, 228 

Fagnani's  Theorem, 229 

The  Hyperbola, 231 

Landen's  Theorem,         . 232 

Graves's  Theorem, 234 

Difference  between  Infinite  Branch  and  Asymptote  for  Hyperbola,  236 

The  Limacon — the  Epitrochoid," 237 

Steiner's  Theorem  on  Rectification  of  Roulettes, 238 

Genocchi's  Theorem  on  Oval  of  Descartes, 239 

Rectification  of  Curves  of  Double  Curvature, 243 

Examples, 246 


V 


£ 


Contents.  xiii 


CHAPTEK  IX. 

VOLUMES   AND    SURFACES   OF   SOLIDS. 

Page 

The  Prism  and  Cylinder, 250 

The  Pyramid  and  Cone, 251 

Surface  and  Volume  of  Sphere, 252 

Surfaces  of  Revolution, 254 

Paraboloid  of  Revolution, .               ■   .  256 

Oblate  and  Prolate  Spheroids, 257 

Surface  of  Spheroid, 257 

Annular  Solids, 261 

Guldin's  Theorems, 262 

Volume  of  Elliptic  Paraboloid, 265 

Volume  of  the  Ellipsoid, 266 

Volume  by  Double  Integration, 269 

Double  Integrals, 273 

Quadrature  on  the  Sphere, 276 

Quadrature  of  Surfaces, 279 

Quadrature  of  the  Paraboloid, 280 

Quadrature  of  the  Ellipsoid, 282 

Integration  over  a  Closed  Surface, 284 

Examples, 288 


CHAPTER  X. 

INTEGRALS     OF     INERTIA 


Moments  and  Products  of  Inertia, 291 

Moments  of  Inertia  for  Parallel  Axes,  or  Planes, 292 

Radius  of  Gyration, 293 

Uniform  Rod  and  Rectangular  Lamina, 294 

Rectangular  Parallelepiped, 295 

Circular  Plate  and  Cylinder, 295 

Right  Cone, 296 

Elliptic  Lamina,      .        . .296 


xiv  Contents. 

Pace 

Sphere, ....  297 

Ellipsoid, 298 

Moments  of  Inertia  of  a  Lamina, 299 

Momental  Ellipse, 300 

Products  of  Inertia  of  a  Lamina, 301 

Triangular  Lamina  and  Prism, 302 

Momental  Ellipse  of  a  Triangular  Lamina, 304 

Tetrahedron, 304 

Solid  Ring, 305 

Principal  Axes, 307 

Ellipsoid  of  Gyration, 309 

Momental  Ellipsoid, 309 

Equimomental  Cone, 310 

Examples, 311 


CHAPTER  XI. 

MULTIPLE     INTEGRALS. 

Double  Integration, 313 

Change  in  Order  of  Integration, 314 

Dirichlet's  Theorem, 316 

Transformation  of  Multiple  Integrals, 321 

Transformation  of  Element  of  a  Surface, 324 

General  Transformation  for  n  Variables, 325 

Green's  Theorems, 326 

Examples, 329 


CHAPTER  XII. 


ON   MEAN   VALUE   AND    PROBABILITY. 


Mean  Values, 333 

Case  of  one  Independent  Variable, 334 

Case  of  two  or  more  Independent  Variables,    .        .        .         .         .        .337 


xv  Contents. 

Page 

Probabilities, 349 

Buffon's  Problem, .         .         .  352 

Curve  of  Frequency, 356 

Errors  of  Observation, 361 

Lines  drawn  at  Random, 368 

Application  of  Probability  to  the  Determination  of  Definite  Integrals,       .  371 

Examples, 374 


Miscellaneous  Examples, 381 


The  beginner  is  recommended  to  omit  the  following  portions  on  the  first 
reading :— Arts.  46,  49,  50,  72-76,  79-81,  89,  96-125,  132,  140,  142-147,  H9> 
158-167,  178,  180,  182,  189-193,  Chapters  x.,  xi.,  xn. 


INTEGKAL    CALCULUS 


CHAPTEE  I. 

ELEMENTARY   FORMS   OF   INTEGRATION. 

i.  Integration. — The  Integral  Calculus  is  the  inverse  of 
the  Differential.  In  the  more  simple  case  to  which  this 
treatise  is  principally  limited,  the  object  of  the  Integral 
Calculus  is  to  find  a  function  of  a  single  variable  when  its 
differential  is  known. 

Let  the  differential  be  represented  by  F  (x)  dx,  then  the 
function  whose  differential  is  F(x)  dx  is  called  its  integral,  and 
is  represented  by  the  notation 

\F(x)dx. 

Thus,  since  in  the  notation  of  the  Differential  Calculus  we 
have 

df(x)=f(x)dx, 

the  integral  of  f'(x)  dx  is  denoted  by  f(x) ;  i.e. 
j/(*)<fc  -/(*), 

Moreover,  as  f{x)  and  f{x)  +  C  (where  C  is  any  arbitrary 
quantity  that  does  not  vary  with  x)  have  the  same  differen- 
tial, it  follows,  that  to  find  the  general  form  of  the  integral  of 
f'{x)  dx  it  is  necessary  to  add  an  arbitrary  constant  to  f(x)  ; 
hence  we  obtain,  as  the  general  expression  for  the  integral 
in  question, 

f/»  <&=/(*)  +  0.  (i) 

[1] 


2  Elementary  Forms  of  Integration. 

In  the  subsequent  integrals  the  constant  C  will  be  omitted, 
as  it  can  always  be  supplied  when  necessary.  In  the  appli- 
cations of  the  Integral  Calculus  the  value  of  the  constant  is 
determined  in  each  case  by  the  data  of  the  problem,  as  will  be 
more  fully  explained  subsequently. 

The  process  of  finding  the  primitive  function  or  the  inte- 
gral of  any  given  differential  is  called  integration. 

The  expression  F(x)  dx  under  the  sign  of  integration  is 
called  an  element  of  the  integral ;  it  is  also,  in  the  limit,  the 
increment  of  the  primitive  function  when  x  is  changed  into 
x  +  dx  (Diff.  Calc,  Art.  7) ;  accordingly,  the  process  of  inte- 
gration may  be  regarded  as  the  finding  the  sum*  of  an  infinite 
number  of  such  elements. 

We  shall  postpone  the  consideration  of  Integration  from 
this  point  of  view,  and  shall  commence  with  the  treatment  of 
Integration  regarded  as  being  the  inverse  of  Differentiation. 

2.  Elementary  Integrals. — A  very  slight  acquaint- 
ance with  the  Differential  Calculus  will  at  once  suggest  the 
integrals  of  many  differentials.  We  commence  with  the 
simplest  cases,  an  arbitrary  constant  being  in  all  cases  under- 
stood.' 

On  referring  to  the  elementary  forms  of  differentiation 
established  in  Chapter  I.  Diff.  Calc.  we  may  write  down  at 
once  the  following  integrals  : — 

f    m  j        ^fl  [dx  -  1 

xmdx  m .  —  = (a) 

J  m  +  1  )xm      (m  -  i)xm~l  v  ' 

(J  =  log(*).  (4) 

f  .  ,         cosmx        C  _       sin  mx 

sin  mxdx  = ,         cos  mx  dx  = .  (c) 

J  m  J  m  w 

— j-  =  tan  x,  -r-^-  =  -  cot  x.  (d) 

J  co§2 a;  J  sm2#  x  ' 


*  It  was  in  this  aspect  that  the  process  of  integration  was  treated  hy  Leib- 
nitz, the  symbol  of  integration  J  being  regarded  as  the  initial  letter  of  the  word 
sum,  in  the  same  way  as  the  symbol  of  differentiation  d  is  the  initial  letter  in 
the  word  difference. 


Fundamental  Forms, 


These,  together  with  two  or  three  additional  forms  which 
shall  be  afterwards  supplied,  are  called  the  fundamental*  or 
elementary  integrals,  to  which  all  other  forms,f  that  admit 
of  integration  in  a  finite  number  of  terms,  are  ultimately  re- 
ducible. 

Many  integrals  are  immediately  reducible  to  one  or  other 
of  these  forms :  a  few  simple  examples  are  given  for  exercise. 

Examples. 

dx  i 

Ans. . 


Cdx 


x 


f   dx 


,  zyx. 

I  tana; dx.  „  -log  (cos a?). 

xn-x  dx  t  i 


[xn-ldx                                            f           i  ,      , 
r      xdx  

I 

i 


sin  9< 


secfl. 


eaxdx.  „  —  ea*. 


*  The  fundamental  integrals  are  denoted  in  this  chapter  hy  the  letters  a,  b,  c, 
&c. ;  the  other  formulae  hy  numerals  i,  2,  3,  &c. 

t  By  integrahle  forms  are  here  understood  those  contained  in  the  elementary 
portion  of  the  Integral  Calculus  as  involving  the  ordinary  transcendental  func- 
tions only,  and  excluding  what  are  styled  Elliptic  and  Hyper- Elliptic  functions. 

[la] 


9. 


Elementary 

For 

ms  of  Integration. 

cdx 
J? 

Ana.  -i. 

X* 

r  dx 

- 

J  xn 

„     wzn. 

f    dx 

)  x-a 

„    log(s-a). 

3.  Integral  of  a  Sum. — It  follows  immediately  from 
Art.  12,  Diff.  Calc,  that  the  integral  of  the  sum  of  any  number 
of  differentials  is  the  sum  of  the  integrals  of  each  taken  sepa- 
rately.    For  example — 

j{Axm  +  Bxn  +  Cjf  +  &c.)  dx=A  jxm  dx  +  Bj  xndx  +  C  \xrdx+&a. 

Axm"     Bx"*1      Cx**1      . 

= + + +  &c.  (2) 

m+in+ir+i  w 

Hence  we  can  write  down  immediately  the  integral  of  any 
function  which  is  reducible  to  a  finite  number  of  terms  con- 
sisting of  powers  of  x  multiplied  by  constant  coefficients. 
Again,  to  find  the  integrals  of  cos2xdx  and  eitfxdx;  here 

f      ,     _      f  1  +  cos  2x  _       x     sin  2a?  ,  . 

cos2xdx  =    dx  =  -  + ,  (3) 


Bm*xdx  =    - 


COS  2X  x      sm  2X  ,   , 

dX   m .  (4) 


2  24 

A  few  examples  are  added  for  practice. 
Examples. 

f  (I  -x^dx  x* 

1.  - ■ .  Ans.  log  x  -  x1  +  — 

J  *  4 

C(x-2)dx  >r.-±- 

3.       /  \&u2xdx  =  J  (sec3  £  -  1)  dx.  „    tan  a;  -  #. 


Integration  by  Substitution. 

sin  Cm  +  n)  x      sin  Cm  —  n)  x 

4.       f  cos mx  cosnxdx.  Ans.  — ; —  H ■ ; — . 

J  2  (m  +  ri)  2  (m  -  n) 

sin  (m  —  n)  x      sin  (m  +  n)x 

c.       f  sin  mx  amnxdx.  ,,     -. r -. —  . 

J        J  2(m-n)  2(m  +  n) 

6.  I  .  / dx.  „     a  sin-1 \/«2  -  #*• 

J  \  «  —  x  a 

Multiply  the  numerator  and  denominator  by  y  a  +  x. 

7.  j  x  \/x  +  a  dx.  Ans.  -  (x  +  af a  (x  4-  af. 

8.  — ■         — .  „     —  ( (x  +  a)  -  x  ) . 

Multiply  the  numerator  and  denominator  by  the  complementary  surd 
V  x  +  a  —  yx. 

C  a  +  bx  .       bx      ab'  -  ba '        .  , 

9.  I  - — —  dx.  Ans.  —  +  — — —  log  (a  +  b'x). 
]  a  +  bx  b  bz 

_  a  +  bx       b         ab'  —  ba' 

Here  — — —  =  77  +  tttt 


a'  +  b'x      V      b'  (a'  +  b'x)' 

4.  Integration  toy  Substitution. — The  integration  of 
many  expressions  is  immediately  reducible  to  the  elementary 
forms  in  Art.  2,  by  the  substitution  of  a  new  variable. 

For  example,  to  integrate  (a  +  bx)n  dx,  we  substitute  z  for 
a  +  bx;  then  dz  =  bdx,  and 

(a+bx)ndx  =  \  —  =  - —  =  v-7 '-— . 

J  v  '  J     b        (n  +  1)  b        (n  +  i)b 

Again,  to  find 


J 


x2dx 


(a  +  bx)nt 
we  substitute  z  for  a  +  bx,  as  before,  when  the  integral  be- 


1  f  (2  -  #)2i 


* 


gn  » 

2a 


b  \{n  -  3)s^3      (n  -  2)z1l~2  +  (n  -  i)^"1)* 


6  Elementary  Forms  of  Integration. 

On  replacing  z  by  a  +  bx  the  required  integral  can  be  ex- 
pressed in  terms  of  x. 

The  more  general  integral 

f    xmdx 


(a  +  bx)ni 


where  m  is  any  positive  integer,  by  a  like  substitution  be- 
comes 

i    C(z-a)mdz 


Expanding  by  the  binomial  theorem  and  integrating  each 
term  separately  the  required  integral  can  be  immediately 
obtained. 

Again,  to  find 

f dx__ 

}xm(a  +  bx)n' 

we  substitute  2  for  -  +  b.  and  it  becomes 
x 

1      Hz-b)mHl-2dz. 

which  is  integrable,  as  before,  whenever  m  +  n  is  a  positive 
eater  than  unity, 
for  example,  we  have 

f       dx       _  1  .      /     x     \ 

J  x  {a  +  bx)      a     °\a  +  bx) 


integer  greater  than  unity 
Thus,  " 


It  may  be  observed  that  all  fractional  expressions  in  which 
the  numerator  is  the  differential  of  the  denominator  can  be 
immediately  integrated. 

For  we  obviously  have,  from  (b), 


)  a  -it- 


Integration  of— — 


Examples. 
sin  x  dx  J  log  (a+b  cos  x) 

Ans. s ; -. 


b  cos  a; 

-££=.  „^-.(*)4. 

J  */as  _  #8  4  W 


3- 


>v/a8  -  #8 
jlog*-. 


J  a?  log  a;* 


f     s2<& 
5*       J  (a  +  h*W 


log: 
i 


6.        f_^_ 


(a  +  bxY 
xdx 


»    2(log*)2- 

„     log  (log  x). 

log  (a  +  fo)  _    3^2  +  4a5a; 
M            fl»           '  2P{a  +  to)2' 

25  ,      a  +  bx 

"  sl0«  — 

«  +  2bx 
a2x  («  +  bx)' 

2  (a  +  fo;)^ 
3*2 

ia  (a  +  3ip)> 
£2 

3  (a  +  bx)% 
5*2 

30  (a  +  &r)i 

2$2          ' 

7'     J  5T+  &*)»" 

8         f     g<fa 
*        J  (a  +  to)*' 

C          dx                                               2             I2X  -  a 
9.  — ;  „    -tan-i      . 

Assume  2«#  -  a2  =  22,  then  «d#  =  zdz,  and  the  transformed  integral  is 

f    idz 
J  a2  +  «*" 


5.  Integration  of    — 

#2  -  a 


Since 


1_  .  _L  f  _■____!_! 

-a2      2«  fa?  -  «     x  -\-  a\' 


(     dx  \   ,      x-  a  ,_. 

weget  J*— a;  =  ^l0sw  w 

This  is  to  be  regarded  as  another  fundamental  formula 
additional  to  those  contained  in  Art.  2. 


8  Integration  of 


a  +  zbx  +  ex2' 
In  like  manner,  since 


i  i 


(x-a){x-(3)      a-P\x-a      x  -  /3j' 

,  [  dx  i      .      x  -  a 

we  have         j  7 r — -^  = log 3.  6) 

J  (a-  -  a)  (x  -  |3)     a  -  j3     &  x  -  )3  w 

Examples. 

C     dx  t  .      <r  -  3 

1.  -5 .  Am.  -  log -. 

Js2-9  6     6  z  +  3 

f  <te  1        x- 3 

2'       ]<*+*)(•  -3)*  "    5     g^T2- 

f  <fa  lo    *  +  4 

J  #»  +  9#  +  20'  "  *  +  5' 

J*       3  2^/3         *+v"3 

6.  Integration  of     \ -. 

a  +  zbx  +  ex1 

This  may  be  written  in  the  form 
cdx 


(ex  +  b)2  +  ac-b2' 
or,  substituting  z  for  ex  +  b, 

dz 

z2  +  ae-  b2' 

This  is  of  the  form  (/)  or  (h)  according  as  ac  -  b2  is  positive 
or  negative. 

[ence,  if  ac  >  b2  we  have 

dx                    1               ,    ex  +  b  ,  v 

tan       r r.'  (7) 


+  2bx  +  ex2     yaV^b2  <Jae  -  b2 


Integration  of — - — -r—1- — ;.  9 

a  +  zbx  +  ex2 


If  ac  <  b2, 


dx                      i          _ .    ex  +  b  -  *Jb2  -  ac       ,_x 
lo% : 7t^=-      (8) 


a  +  2bx  +  ex2      2*/b2-ac        cx  +  b  +  */b2 


ae 


This  latter  form  can  be  also  immediately  obtained  from  (6). 
In  the  particular  case  when  ac  =  b2,  the  value  of  the  inte- 
gral is 

-  i 


7.  Integration  of 


ex  +  b' 
(p  +  qx)  dx 


a  +  2bx  +  ex2 

This  can  at  once  be  written  in  the  form 

q    (b  +  ex)  dx       pc  -  qb  dx 

c  a  +  ibx  +  ex2  c      a  +  2bx  +  ex2* 

The  integral  of  the  first  term  is  evidently 

—  log  (a  +  2bx  +  ex2), 

while  the  integral  of  the  second  is  obtained  by  the  preceding 
Article. 

For  example,  let  it  be  proposed  to  integrate 

(x  cos  0  -  i)dx 
x2-2xcos9+  1" 

The  expression  becomes  in  this  case 

cos  0  (x  -  cos  6)  dx  sin2  Odx 


x2  -  2x  cos  0  +  1       (x  -  cos  0)2  +  sin2  6 9 
hence 

((x  cos  6  -  1)  dx      cos  9  , 
-3 5 = log  (x2  -  2X  COS  6  +  I) 
X2  -  2X  COS  B  +  I             2  &  V  ' 

•    a  l       1 x  -  cos  0  /  \ 

-sin&  tan-1  — .    -    .  (o) 

sin  v  v  ' 


10  Elementary  Forms  of  Integration. 

When  the  roots  of  a  +  2bx  +  ex2  are  real,  it  will  he  found 
simpler  to  integrate  the  expression  by  its  decomposition  into 
partial  fractions.  A  general  discussion  of  this  method  will 
be  given  in  the  next  chapter. 


Examples. 
I r.  Ana.  —yz  tair1  [  — —  ) . 

J  i  +  *  +  *«  ,/j       \  tyl  J 

I  STMTS'  -    tan-1(,  +  2). 


4* +  5 

f  dx r         i^  +  a 

5'       J  5** +4*+ 8*  "    6tan     "e-* 

f  x2dx  i  .       / 1  +  «3\ 

8.        ( ^— -.  „    W(2*-i). 

J    I   -  2X  +  2X* 

8.  Exponential  Value  for  sin  0  and  cos  0. — By  com- 
paring the  fundamental  formulae  (/)  and  (h)  the  well-known 
exponential  forms  for  sin  0  and  cos  0  can  he  immediately 
deduced,  as  follows : 

Substitute  z  */-  i  f or  #  in  both  sides  of  the  equation 

and  we  get 

f    dz  _J ,      /i  +sv/:~i\  , 

=   7=  log  [  7=r      +  C0WS£./ 


Exponential  Forms  of  sin  9  and  cos  6>  11 

or,  by  (/),     tan"1  z  =       '_  iog I *  *  *     _J  J  +  const. 

Now,  let  z  =  tan  0,  and  this  becomes 

„  i       ,      (\  +  V7"-  i  tan0\ 

0  =  — 7=.  Jog  I  =: +  const. 

2*/-  i         \i  -  v-  i  tan  0/ 

When  6  =  o,  this  reduces  to  o  -  cowstf. 

xt  .«r;     cos  0  +  a/-  i  sin  0      ,       .        / —    .    „. ,  ■ 

Hence     e2^-1  = ; —  =  (cos  0  +  V-  i  sin  0)2, 

cos  6  -  v  -  i  sin  6 

or  e^  =  cos  0  +  */-  i  sin  0, 

e-H-i  =  Cos  0  -  v^-  i  sin  0. 

dx 
g.  Integration  of 


*/x"  ±  a2 
Assume*  ^/x%  ±  a2  =  z  -  x, 

then  we  get  ±  a2  =  s2  -  2xz, 

hence  (2  -  x)  dz  =  zdx.  or =  — ; 

'  Z  -  X         z 

f      dx  rjz  

■•   J7^T^=Ji  =  log^  =  log(^+^2±a2)-    w 

This  is  to  be  regarded  as  another  fundamental  form. 

By  aid  of  this  and  of  form  (e)  it  is  evident  that  all  ex- 
pressions of  the  shape 

dx 

*/a  +  zbx  +  ex2 


*  The  student  will  better  understand  the  propriety  of  this  assumption  after 
reading  a  subsequent  chapter,  in  which  a  general  transformation,  of  which  the 
above  is  a  particular  case,  will  be  given. 


12  Elementary  Forms  of  Integration, 

can  be  immediately  integrated ;  a,  b,  c,  being  any  constants, 
positive  or  negative. 

The  preceding  integration  evidently  depends  on  formula 
(t),  or  (e),  according  as  the  coefficient  of  x2  is  positive  or 
negative. 

Thus,  we  have 

,  .  ;  =  ~~r  log  (  ex  +  b  +  yTCa  +  zbx  +  ex2)  ,  (10) 

)y/a+2bx  +  cx2     */c        \  7 

r  dx  i      .    ,  /  ex  -  b   \  ,     x 

■  =  —  sm-1  ],  (il) 

J  *fa  +  zbx  -  ex1     */c  \^/ac  +  bv 

c  being  regarded  as  positive  in  both  integrals. 

When  the  factors  in  the  quadratic  a  +  ibx  +  ex2  are  real, 
and  given,  the  preceding  integral  can  be  exhibited  in  a 
simpler  form  by  the  method  of  the  two  next  Articles. 

dx 
10.  Integration  of 


v/(«-a)(«-^) 


Assume  x  -  a  =  s2,     then  dx  =  2zd%  ; 

dx 

=  2dz  ; 


hence 


^/x  -  a 
dx 


*/(x-a)(x-$)       ^/32  +  a-j3, 
dx f         dz 

yo^oF3^)  ~  2J  yz*+a-p 


=  2  log  (*  +  V?  +  a  -  0),  by  (t), 


or 


f  dx  / 

bfa-a^-ar210^-^^7^     (I2) 


Exponential  Forms  of  sin  9  and  cos  6.  13 

dx 


ii.  Integration  of 


\/>-a)(j3-a?) 

As  before,  assume  x  -  a  =  z2,  and  we  get 
dx  idz 


V(x  -  a)  (j3  -  X)      V$-a-  s2' 
Hence,  by  (e), 

dx  .        \x  -  a 

—T  =  2  sm   J73 • 

W(x-a)(P-x)  VP-a 

Otherwise,  thus : 

assume        x  =  a  cos2  0  +  j5  sin2  0, 

then  /3  -  x  =  (j3  -  a)  cos20,     a?  -  a  =  (j3  -  a)  sin2 0, 

and  dfe  =  2(j3  -  a)  sin  0  cos  6  dd ; 

hence  r — =  zdO ; 


13) 


20=2  sin 


1    /a?  -  a 


JA-«)(/3-^)  V/3 

12.  Again,  as  in  Art.  7,  the  expression 

(p  +  qx)  dx 
va  +  ibx  +  cxz 
can  be  transformed  into 

q      (b  +  c#)  dx         pc  -  qb  dx 

c  ^/a  +  2bx  +  ex*  c      ^/a  +  ibx  +  ex* 

and  is,  accordingly,  immediately  integrable  by  aid  of  the 
preceding  formulae. 


14  Elementary  Forms  of  Integration. 

Examples. 
f      dx  


\/az  - 


'•  """J;- 


5*       J  J^ — 7  rf*  =  vA*  +  «)(*  +  *)  +  («-*)  log  {\/x  +  a  +•  \/a;  +  b). 
Multiply  the  numerator  and  denominator  by  ^/x  +  a. 
f  dx 

'      W7^x~ 


3"  /  .   =•  »     asin-iv/*-i. 

J  V  l*  -  x2  -  2 

4-        [-7==..  „     log  (2a;  +  1  +  2\/l  +  s  +&). 

J  v  1  +  x  +  %• 


d*  .  .     .  2X+ 

°-       1  -  ,  — .  Ans.  sin"1 


v/i-s-x2  ^5 


8.  Show,  as  in  Art.  8,.  by  comparing  the  fundamental  formulae  (e)  and  (i), 
that 


13.  Integration  of 

Let  x  -  p  =  -,  then 


0  +  ^/_  1  sin  0  =  e6**. 

dx 


(x  - p)  V  a  +  ibx  +  ex7. 


dx  dz       ,         1  + 

and  x  = 


a?  -  p         z  z 

f dx f     -  dz 

J  (x  -p)*/a  +  2bx  +  ex2     J  ^/azz+  2bz(i  +pz)  +c(i  +pz)2 

f  dz 

J  yd  +  20'z  +  cfz2 ' 
where  a'  =  c,     b'  =  b  +  cp,     c'  =  a  +  ibp  +  cp2. 

The  integral  consequently  is  reducible  to  (10),  or  (11),  ac- 
cording as  c  is  positive  or  negative. 


dx 


V*8  —  a 


dx 
Integration  of  -. — — — .  15 

"  {a  +  ex2)* 


Examples. 

Ans.  -  cos"1  (  -  ) 
a  \xj 


f         dx  ,       /a/  i  +  x%  -  i\ 

J  x^/x*  +  I  V  *  / 

r  dx  Jl  —  X 

4-  I — .     Ans.  — —  log  (  jr —  — -  1 . 

J  x\/ a  +  2bx  +  ex2  *y  a         \a  +  bx  +  \/ «y  a+2bx  +  cx  I 

f  dx  M  \       .        I      bx  —  a      \ 

5-  — ,  =•  -4»*.  ~tt.  sin-i  (  —  ) . 
J  ^V  c#2  +  2##  -  a                                V"            \xyac+b2' 

A  f  ^  '  •      1    l*\/~Z\ 

6. -.  „     — —   sm-1  I  ■ ) . 

J  (i  +  a;)\/i  +  2^ -ic2  \/2  V1^^/ 

7-  .  „     sia-i  ( —  J. 

J  (i  +  x)*/ 1  +x-x*  \(i+x)VS' 

14.  The  transformation  adopted  in  the  last  Article  is  one 
of  frequent  application  in  Integration.  It  is,  accordingly, 
worthy  of  the  student's  notice  that  when  we  change  x  into 

1  1         dx        dz          ..   .                _    ..          1    dx        dz 
-  we  nave  —  = ;  and,  m  general,  11  #  =  -,  —  ■ • 

2  x  z  °         •  z    x         nz 

These  results  follow  immediately  from  logarithmic  differ- 
entiation, and  often  furnish  a  clue  as  to  when  an  Integration 
is  facilitated  by  such  a  transformation. 

For  example,  let  us  take  the  integral 


I 


dx 


x(a  +  bxn)' 


Here,  the  substitution  of  -  for  xn  gives 


1  f    dz 
n  J  az  + 


16  Elementary  Forms  of  Integration. 

The  value  of  which  is  obviously 

log  (az  +  b)y  or  —  log( —  )• 

na     &  v  n        na     °\a  +  bxnJ 

Again,  to  integrate 


dx 


x  y/ax"  +  b 

assume  xn  =  -,  and  the  transformed  integral  is 

2(       dz 
»  J  </~a  +  bi2 

This  is  found  by  (e)  or  (i)  according  as  b  is  positive  or 
negative. 

dx 

15.  Integration  of      -. -r.. 

0  (a  +  cx2)$ 

Let  #  «  -  and  the  expression  becomes 

zdz 


(az*  +  c)%  * 
the  integral  of  this  is  evidently 


1  x 

a  or 


a  (az2  +  c)%        a  (a  +  cx2)V 

Hence  -. -jr.  =  -7 nr%.  (14) 

J  (a  +  as2)*     a  (a  +  c#2)* 

16.  To  find  the  integral  of 

dx 

(a  +  ibx  +  ex2)*' 

This  can  be  written  in  the  form 

c^dx 


{ac  -  b*  +  (ex  +  by}*' 
which  is  reduced  to  the  preceding  on  making  ex  +  b  =  z. 


Integration  of  - — h.  17 

sm  9 

Hence,  we  get 

f dx b  +  ex 

J  {a  +  2bx  +  cx2f  "  {ac-  b2) {a  +  ibx  +  cx2)^'  ^5' 

Again,  if  we  substitute  -  for  x, 

z 

xdx 


becomes 


(a  +  2bx  +  cx2)%  {az2  +  2bz  +  c)$ 

and,  accordingly,  we  have 

f  xdx  a  +  bx 


J  (a  +  2bx  +  ex2)*        {ac  -  b2)  {a  +  2bx  +  ex2)* 

Combining  these  two  results,  we  get 

f     {p  +  qx)  dx  bp  -  aq  +  (cp  -  bq)  x 

J  {a  +  2bx  +  ex2)*      {ac  -  b2){a  +  2bx  +  ex2)* 

_   M              dO 
17.  Integration  of  - — ■*  and -„. 

sm  9  cos  9 


:«6) 


It  will  be  shown  in  a  subsequent  chapter  that  the  integra- 
tion of  a  numerous  class  of  expressions  is  reducible  either  to 

that  of  - — 7j,  or  of 7, :  we  accordingly  propose  to  inves- 

sm  0'  cos  9  &  J  r    * 

tigate  their  values  here.     For  this  purpose  we  shall  first  find 

the  integral  of  -r— s ^ 

0  sm  9  cos  9 

dO 


H  d9  cos2fl     ^(tanfl). 

sin  9  cos  9     tan  9        tan  9    9 

-r-Tj 7j  =  log  (tan  0) .  (17) 

sm  9  cos  9       ° 

M 


18  Elementary  Forms  of  Integration, 

Next,  to  find  the  integral  of 
dO 


sin  0' 
This  can  be  written  in  the  form 
eg 

.  e     e9 

2  sin- cos - 

2  2 

and,  by  the  preceding,  we  have 

JA:*K>  <■•> 

Again,  to  determine  the  integral  of  — ^  we  substitute 

-  -  (f)  for  0,  and  the  expression  becomes  -r-^  :   the  integral 
of  this,  by  ( 1 8),  is 

-  log  I tan  | J,  or  log  (cot  ^\  or  log  jcot  f  ^  -  -Jj . 
Accordingly,  we  have 

fJ»-*Krf-*M;*I))-  <"> 

This  integral  can  also  be  easily  obtained  otherwise,  as 
follows : — 

f  dO    _  r  cos  OdO  _  r^(sin0) 
J  COS0     J    cos20        J    cos20 

Let  sin  0  =  x,  and  the  integral  becomes 

Jdx        i .      fi  +  x\      i  .      (\  +  sin  0\ 

The  student  will  find  no  difficulty  in  identifying  this 
result  with  that  contained  in  (19). 


Integration  of  — ; 5  19 

a  +  0  cos  ft 


18.  Integration  of 


#  +  6  cos  0' 

This  can  be  immediately  written  in  the  form 
£0 

(a  +  b)  cos3  -  +  (a  -  b)  sin2  - 

sec2-  <#0 
2 
or  


a  +  b  +  (a  -  b)  tan2  - 
on  substituting  z  for  tan  -  this  beoomes 


a  +  b  +  (a  -  b)zz 
Consequently,  by  Ex.  6,  Art.  2,  we  get 

(1)  when  a  >  b, 

)7Tb^re  -  y=frp tan_1 1  fel  tan!j-      (2°) 

(2)  when  0  <  5,  by  formula  (h), 

r        JQ  j  I  </b  +  a  +  </b-atfm-  I 

J.-TTc^-rTp^10^  , ,— — ef'{21) 

*/b  +  a  -  v  #  -  «  tan  - 

If  we  assume  a  =  b  cos  a,  we  deduce  immediately  from 
the  latter  integral 

f        a-B 

f    jg        I  10  Jcos~ 

J  cos  a  +  cos  0      sin  a    °^   ]         a+  0 

cos 

L  2 

The  integral  in  (20)  can  be  transformed  into 
f        d9  1  x  lb  +  a  cos  0) 

)a  +  b  cos  0~  v/^~f*  C°S    (a+&cos0J* 

[2  a] 


20  Elementary  Forms  of  Integration. 

In  a  subsequent  chapter  a  more  general  class  of  integrals 
which  depend  on  the  preceding  will  be  discussed. 

19.  Methods  of  Integration. — The  reduction  of  the 
integration  of  functions  to  one  or  other  of  the  fundamental 
formulae  is  usually  effected  by  one  of  the  following  methods : — 

(1).  Transformation  by  the  introduction  of  a  new  va- 
riable. 

(2).  Integration  by  parts. 

(3).  Integration  by  rationalization. 

(4).  Successive  reduction. 

(5).  Decomposition  into  partial  fractions. 

Two  or  more  of  these  methods  can  often  be  combined 
with  advantage.  It  may  also  be  observed  that  these  different 
methods  are  not  essentially  distinct :  thus  the  method  of 
rationalization  is  a  case  of  the  first  method,  as  it  is  always 
effected  by  the  substitution  of  a  new  variable. 

We  proceed  to  illustrate  these  processes  by  a  few  ele- 
mentary examples,  reserving  their  fuller  treatment  for  sub- 
sequent consideration. 

20.  Integration  by  Transformation. — Examples  of 
this  method  have  been  already  given  in  Arts.  4,  10,  &c.  One 
or  two  more  cases  are  here  added. 

Ex.  1.  To  find  the  integral  of  sin2  a;  ao&xdx. 
Let  sin  x  =  y,  and  the  transformed  integral  is 


Jif^-W*'-J/*-{»'*-j-5--X" 


surx 


Ex.  2. 

h 

e*dx 
:  +  e2*' 

Lete*  = 

■■'!/, 

and  we 

get 

f  dy 

-fa 

xflv  = 

J 1  +  y* 

21.  Integration  by  Parts. — We  have  seen  in  Art.  13, 
Diff.  Calc,  that 

d(uv)  =  udv  +  vdu; 
hence  we  get 


or 


uv  =  j  udv  + 1  vdu, 
(udv  =  uv  -  J  vdu.  (22) 


Integration  by  Parts.  21 

Consequently  the  integration  of  an  expression  of  the  form 
udv  can  always  be  made  to  depend  on  that  of  the  expression 
vdu. 

The  advantage  of  this  method  will  be  best  exhibited  by 
applying  it  to  a  few  elementary  cases. 

xdx 


n  f  •    ,    j  •    .        f     xdx 

Ex.  I.  em'1  xdx  =  a?sin_1#  - 

J  J  <yi  ~  %' 


=  x  sin-1#  +  <Si  -  x2. 


Ex.  2.  x  log  xdx. 

x2 
Let  u  =  log  x,  v  =  — ,  and  we  get 

f    .  .       x2\6gx      i  f  ,<fo     a?Vi  x\ 

j  x  log  *  dx  =  — -S-  -  -  j  ^  _  =  -  (tog  «  -  -J. 

Ex.  3.  eaxxdx. 

eax 
Let  a;  p  u,  —  =  0,  then 

a 

I  #ea*d#  = —  dx=  —  [x . 

J  a       ]  a  a  \        a) 

Ex.  4.  I  eax  sin  mx  dx. 

eax 
Let  sin  mx  =  u.  —  =  v.  then 

a 

Jn/r   .          ,       e°*  sin  mx     mCm 
e™  sin  m#  dx  = eoa;  cos  mxdx. 
a             a) 

0.    .,    ,          |*                  _       e°*cosm#     #?  f  „„  . 
Similarly,        e°*  cos  mx  dx  = +  —  \eax  sin  w#  dk. 


22  Elementary  Forms  of  Integration. 

Substituting,  and  solving  for  J  eax  sin  mxdx,  we  obtain 

f  „,  .  ,       eax(a  sin  mx  -  m  cos  mx)  ,     „ 

e"x  sin  wacfo  =  — i '-.  (23) 

J  a2  +  m2  v     ' 

In  like  manner  we  get 

f  „  ,       «"*  (a  cos  mx  +  m  sin  w#)  .     . 

eax  cos  *w#  dx  =  — s = \  (24) 

J  a2  +  m2  v  ^7 

Ex.  5.  ^a2  +  xtdx. 

Let  \/a2  +  #2  =  w,  then 

v^tf2  +  a?  dx  =  #  \/a2  +  #2  -1     ;  ; 

J  J  yV  +  ? 

also      fy^2^  =  a2f^JL,J^=. 

Hence,  by  addition,  and  dividing  by  2, 

\</77*dx  =  "^  +  ^  +  £  log(^+  v/^T^).  (25) 

Ex.  6.  J  i0g  (^  +  yx*  ±  a»j  ^ 

Here    J  i0g  (^  +  y^jji  ±  a2)  ^  =  x  \0g  (%  +  y/^  ±  ^ 

yV  ±  a2 
=  x  log  (a?  +  \/x2  ±  a2)  -  \Zx2  ±  a2.  (26) 


Integration  by  Rationalization.  23 


Examples. 

I. 

1  xnlogxdx. 

,4ws.  [log a; ). 

2. 

1  tan_1a?*?a\ 

„    a:  tan-1  a;  —  log(i+a;2). 

3- 

1  x  tan2  xdx. 

X2 

,,    a:  tan  a;  +  log  (cos  x) . 

4- 

J  (i  -  *»)*' 

'x  sin-1  a;       i                   ^ 

»        / z  +  .,  iog  v1      a  /• 

Let 

a;  =  sin  y,  and  the 

integral  becomes 

1^^=^ 

(?(tan 

y)  =  y  tan  2/ 

'■+  log  (cos  2/). 

5- 

i  e*x2dx. 

„      tf*  (X2  -  2X  +  2). 

22.  Integration  by  Rationalization. — By  a  proper 
assumption  of  a  new  variable  we  can,  in  many  cases,  change 
an  irrational  expression  into  a  rational  one,  and  thus  inte- 
grate it.  An  instance  of  this  method  has  been  given  in 
Art.  8. 

The  simplest  case  is  where  the  quantity  under  the  radical 
sign  is  of  the  form  a  +  bx  :  such  expressions  admit  of  being 
easily  integrated. 

For  example,  let  the  expression  be  of  the  form 

xndx 


(a  +  bx)V 

where  n  is  a  positive  integer.     Suppose  a  +  bx  =  s2,  then 

izdz 
ax  =  — — ,  and  x 


b    ' b 

making  these  substitutions,  the  expression  becomes 
2(s2  -a)ndz 


24  Elementary  Forms  of  Integration. 

Expanding  by  the  Binomial  Theorem  and  integrating  the 

terms  separately,  the  required  integral  can  be  immediately 

xn  dx 
found.     It  is  also  evident  that  the  expression can 

(a  +  bx)t 
be  integrated  by  a  similar  substitution. 

q$m+i  fa 

23.  Integration  of        ■ 

\a  +  carp 

where  m  is  a  positive  integer. 

Let  a  +  ex7,  =  z2 ;  then  xdx  =  — ,  x7,  = ;  and  the 

c  c 

transformed  expression  is 

(z*  -  a)mdz 

This  can  be  integrated  as  before.     It  can  be  easily  seen 

/g2m+l  fa 

that  the  expression is  immediately  integrable  by 

(a  +  ex2)* 
the  same  substitution. 

A  considerable  number  of  integrals  will  be  found  to  be 
reducible  to  this  form  :  a  few  examples  are  given  for  illustra- 
tion. 

Examples. 

f     &**  a      (l  -  *2)*    / 

1.  1  —         .  Am. '-  -  (1  -  x2)i. 

f      afidx  z6      2z3  ,  , -« 

2.  I  ——=.  „ +  z  ;  where  z  =  y  1  +  x2. 

Jyi+x*  5        3 

c     a?dx  -  {2a  +  $cx*) 

J  (a  +  afifi  "     3c2  (a  +  cxrf' 

24.  It  is  easily  seen  that  the  more  general  expression 

f(x2)  xdx 


</a* 


+  cx" 


where  f(x2)  is  a  rational  algebraic  function,  can  be  ration- 
alized by  the  same  transformation. 


Integration  of  1—A — -j——. -?.  25 

9  ( A  +  Cx2)  (a  +  cx2)* 


Again,  if  we  make  x  =  -  the  expression 

dx 


xn  (a  +  cx2)* 

transforms  into 

zn^dz 

and  is  reducible  to  the  preceding  form  when  n  is  an  even  posi- 
tive integer. 

Hence,  in  this  case,  the  expression  can  be  easily  integrated 
by  the  substitution  [a  +  cx2)*  =  xy. 

It  will  be  subsequently  seen  that  the  integrals  discussed 
in  this  and  the  preceding  Articles  are  cases  of  a  more  general 
form,  which  is  integrable  by  a  similar  transformation. 

Examples. 


f         &x  a         V*   - J    /     •>        N 

T7i — u-  Am-  ~ — ; —  (2X  +  0- 


y/x2- 

^ (*2+i)*  (Q      4    , 

I  +  x2)i' 

25.  Integration  of 


f         dx 

J  X*  (I  +  X 

dx 


2)r  "      i5x    r  x*  + 


{A+  Cfc»)  (a* +«&*)*" 

As  in  the  preceding  Article,  let  (a  +  cx2)*  =  xz,  or 
a  +  cx2  =  x2z2 :  then,  if  we  differentiate  and  divide  by  2x,  we 
shall  have 

j        ,j            7         dx       dz 
cdx  =  z2dx  +  xz  dz.  or  —  = 1, 


xz      c  -  z* 
dx 


'  (a  +  cx2)*      c-z2' 
and  the  transformed  expression  evidently  is 


(27) 


(Ac'-  Co)  -  Az* 


26  Elementary  Forms  of  Integration. 

This  is  reduoible  to  the  fundamental  formula  (ti),  or  (/), 

,.           Ac  -  Ca  . 
according  as -. —  is  positive  or  negative. 

Hence,  (i)  if ^ >  °>  the  integral  is  easily  seen  to  be 

T  .      (*/ Aia  +  ex2)  +  x^/Ac  -  Ca\       .  n, 

lQg         '        i 7==F  •    (28) 


2v/^  (^c  -  Ca)     &  \</A{a  +  ex2)  -  x^ Ac  -  Ca 
(2).  If —  <  o,  the  value  of  the  integral  is 


.  x^/Ca  -Ac  .     x 

tan"1  -7 —  (29) 


v^-4  ( C«  -  Ac)  *yA  (a  +  ex2) 

Examples. 

[  dx  r  /         52;         \ 

2'       J  (3  +  4*<)  (4  -3*2)>'  "    77i  VTT^^-J- 

f <& 2.  1     2a/3  +  4*2  +  5* 

3'       J  (4 -3^(3  +  4^*  M     2Ol0g2^/3T^_5a;- 

26.  Rationalization  by  Trigonometrical  Trans- 
formation.— It  can  be  easily  seen,  as  in  Art.  6,  that  the 
irrational  expression  ^/ a  +  zbx  +  ex2  can  be  always  trans- 
formed into  one  or  other  of  the  following  shapes: 

(I)    («*-«•)»,      (2)  (a* +  **)!,      (3)(**-a>)i; 

neglecting  a  constant  multiplier  in  each  case. 

Accordingly,  any  algebraic  expression  in  x  which  con- 
tains one,  and  but  one,  surd  of  a  quadratic  form,  is  capable 
of  being  rationalized  by  a  trigonometrical  transformation  : 
the  first  of  the  forms,  by  making  z  =  a  sin  0 ;  the  second,  by 
z  =  a  tan  0  ;  and  the  third,  by  z  =  a  sec  0. 


Rationalization  by  Trigonometrical  Transformation.      27 

For,  (i)  when  z  =  a  sin  0,  we  have  (a2  -  s2)^  =  a  cos  9,  and 
ofe  =  a  cos  0^0. 

(2).  When  s  =  a  tan  0,  ....  (a2  +  s2)*  =  a  sec  0,  and 
adO 
cos*5*/ 

(3).  When  2  =  a  sec  9,  ....  (z2  -  a~)*  =  a  tan  0,  and 
afe  =  a  tan  0  seo9d9. 

A  number  of  integrations  can  be  performed  by  aid  of  one 
or  other  of  these  transformations.  In  a  subsequent  place  this 
class  of  transformations  will  be  again  considered.  For  the 
present  we  shall  merely  illustrate  the  method  by  a  few  ex- 
amples. 

Examples. 

dx 


r       dx 

J*2(I  +  s2)i 
e 

J  cos  0  dd  _  c 
sin20    =J 


Let  x  =  tan  0,  and  the  integral  becomes 

cos  0  dd      C  d (sin  0)  I  \/  I 


sinz0  sin  0 

f        <& 

J  (a3  -  &f 
Let  a;  =  a  sin  0,  and  we  get 

dd       tan 


f     tf0    _  tan  0  _ 
J  cos20  ™  "a2-  7 


This  has  heen  integrated  by  another  transformation  in  Art  15. 

dx 

1*' 

Let  a;  =  sec  0,  and  the  integral  becomes 


3-  f        dx 


f       <>  „  7„           -l     /  >  .           sin  0  cos  0 
I  cos*  d  dQ  ;  or,  by  (3)  Art.  3,  + 

accordingly,  the  value  of  the  integral  in  question  is 
\/x2  -  1 


2X*         +  2 


28  Elementary  Forms  of  Integration. 


J      ('+*')* 


Let  x  =  tan  0,  and  we  get 

e"0  (a  cos  0  +  sin  0) 


cose^dO]  or  by  (23), 


i+«2 


f  <fa;  c«  tan"1*        (a  +  x)  ea  tAa~1'  * 

Hence  -  = . 

J    (i+s2)3       (1  +  a2)(l+s2)* 

5.  <fc  sin"1  [  — —  )  . 

Let =  sin2  0,  or  x  —  a  tan2  6,  and  the  integral  becomes 

a  +  x 

a^e rf (tan20),  or  a  $6 d (sec20) :  (since  sec20  =  I  +  tan20). 
Integrating  by  parts,  we  liave 

j$d  (sec2  0)  =  0  sec2 0  -  J  sec2 Qdd  =  d  sec2 0  -  tan  0  : 
hence  the  value  of  the  proposed  integral  is 


[a  +  x)  tan" 


It  may  be  observed  that  the  fundamental  formulae  (e)  and  (/)  can  be  at  once 
obtained  by  aid  of  the  transformations  of  this  Article. 

27.  Remarks  on  Integration. — The  student  must 
not,  however,  take  for  granted  that  whenever  one  or  other  of 
the  preceding  transformations  is  applicahle,  it  furnishes  the 
simplest  method  of  integration.  "We  have,  in  Arts.  9  and  13, 
already  met  with  integrals  of  the  class  here  discussed,  and 
have  treated  them  by  other  substitutions:  all  that  can  Be 
stated  is,  that  the  method  given  in  the  preceding  Article  will 
often  be  found  the  most  simple  and  useful.  The  most  suit- 
able transformation  in  each  case  can  only  be  arrived  at  after 
considerable  practice  and  familiarity  with  the  results  intro- 
duced by  such  transformations. 

By  employing  different  methods  we  often  obtain  integrals 
of  the  same  expression  which  appear  at  first  sight  not  to 
agree.  On  examination,  however,  it  will  always  be  found 
that  they  only  differ  by  some  constant ;  otherwise,  they  could 
not  have  the  same  differential. 


Observations  on  Fundamental  Forms.  29 

28.  Higher  Transcendental  Functions.— Whenever 

the  expression  under  the  radical  sign  contains  powers  of  x 
beyond  the  second,  the  integral  cannot,  unless  in  exceptional 
cases,  be  reduced  to  any  of  the  fundamental  formulae ;  and 
consequently  cannot  be  represented  in  finite  terms  of  x,  or  of 
the  ordinary  transcendental  functions  :  i.  e.  logarithmic,  ex- 
ponential, trigonometrical,  or  circular  functions.  Accord- 
ingly, the  investigation  of  such  integrals  necessitates  the 
introduction  of  higher  classes  of  transcendental  functions. 

Thus  the  integration  of  irrational  functions  of  cc,  in  which 
the  expression  under  the  square  root  is  of  the  third  or  fourth 
degree  in  x,  depends  on  a  higher  class  of  transcendentals 
called  Elliptic  Functions. 

29.  The  method  of  integration  by  successive  reduction  is 
reserved  for  a  subsequent  place.  The  integration  of  rational 
fractions  by  the  method  of  decomposition  into  partial  frac- 
tions will  be  considered  in  the  next  chapter. 

30.  Observations  on  Fundamental  Forms. — From 
what  has  been  already  stated,  the  sign  of  integration  (J)  may 
be  regarded  in  the  light  of  a  question :  i.  e.  the  meaning  of 
the  expression  j  F(x)  dx  is  the  same  as  asking  what  function 
of  x  has  F(x)  for  its  first  derived.  The  answer  to  this  ques- 
tion can  only  be  derived  from  our  previous  knowledge  of  the 
differential  coefficients  of  the  different  classes  of  functions,  as 
obtained  by  the  aid  of  the  Differential  Calculus.  The  number 
of  fundamental  formulae  of  integration  must  therefore,  ulti- 
mately, be  the  same  as  the  number  of  independent  kinds  of 
functions  in  Algebra  and  Trigonometry.  These  may  be 
briefly  classed  as  follows  : — 

(1).  Ordinary  powers  and  roots,  such  as  xm7  xq,  &c. 

(2) .  Exponentials,  ax,  &c,  and  their  inverse  functions ; 

viz.,  Logarithms. 
(3).  Trigonometric  functions,  sin#,  tan#,  &c,  and  their 

inverse  functions ;  sin"1^,  tan-1#,  &c. 

This  classification  may  assist  the  student  towards  under- 
standing why  an  expression,  in  order  to  be  capable  of  inte- 
gration in  a  finite  form,  in  terms  of  x  and  the  ordinary 
transcendental  functions,  must  be  reducible  by  transforma- 
tion to  one  or  other  of  the  fundamental  formulae  given  in 


30  Elementary  Forms  of  Integration. 

this  chapter.  He  will  also  soon  find  that  the  classes  of  in- 
tegrals which  are  so  reducible  are  very  limited,  and  that  the 
large  majority  of  expressions  can  only  be  integrated  by  the 
aid  of  infinite  series. 

The  student  must  not  expect  to  understand  at  once  the 
reason  for  each  transformation  which  he  finds  given  :  as  he, 
however,  gains  familiarity  with  the  subject  he  will  find  that 
most  of  the  elementary  integrations  which  can  be  performed 
group  themselves  under  a  few  heads ;  and  that  the  proper 
transformations  are  in  general  simple,  not  numerous,  and 
usually  not  difficult  to  arrive  at.  He  must  often  be  prepared 
to  abandon  the  transformations  which  seemed  at  first  sight 
the  most  suitable  :  such  failures  are  not,  however,  to  be  con- 
sidered as  waste  of  time,  for  it  is  by  the  application  of  such 
processes  only  that  the  student  is  enabled  gradually  to  arrive 
at  the  general  principles  according  to  which  integrals  may  be 
classified. 

Many  expressions  will  be  found  to  admit  of  integration 
in  two  or  more  different  ways.  Such  modes  of  arriving  at 
the  same  results  mutually  throw  light  on  each  other,  and  will 
be  found  an  instructive  exercise  for  the  beginner. 

31.  Definite  Integrals. — We  now  proceed  to  a  brief 
consideration  of  the  process  of  integration  regarded  as  a  sum- 
mation, reserving  a  more  complete  discussion  for  a  subsequent 
chapter. 

If  we  suppose  any  magnitude,  u,  to  vary  continuously  by- 
successive  increments,  commencing  with  a  value  a,  and  termi- 
nating with  a  value  /3,  its  total  increment  is  obviously  repre- 
sented by  /3  -  a.  But  this  total  increment  is  equal  to  the  sum 
of  its  partial  increments ;  and  this  holds,  however  small  we 
consider  each  increment  to  be. 

This  result  is  denoted  in  the  case  of  finite  increments  by 
the  equation 

2  (Aw)  =  0  -a; 

a 

and  in  the  case  of  infinitely  small  increments,  by 

(30) 


du  =  (5  -  a ; 


Definite  Integrals.  31 

in  which  |3  and  a  are  called  the  limits  of  integration  :  the 
former  being  the  superior  and  the  latter  the  inferior  limit. 
Now,  suppose  u  to  be  a  function  of  another  variable,  x, 
represented  by  the  equation 

u  =f{x) : 

then,  if  when  x  =  a,  u  becomes  a,  and  when  x  =  b,u  becomes 
j3,  we  have 

Moreover,  in  the  limit,  we  have 

du  =/'(#)  d#, 

neglecting*  infinitely  small  quantities  of  the  second  order 
(See  Diff.  Calc,  Art.  7). 

Hence,  formula  (30)  becomes 


1. 


/(*)  dx  =/(J)  -/(«) ;  (31) 


in  which  5  and  #  are  styled  the  superior  and  the  inferior  limits 
of  a?,  respectively. 


Tt  should  be  observed  that  the  expression     f(x)dx9 


re- 


presents here  the  limit  of  the  sum  denoted  by  S  (/'(#)  A#), 

when  Ax  is  regarded  as  evanescent. 

In  the  preceding  we  assume  that  each  element  f\x)  dx  is 
infinitely  small  for  all  values  of  x  between  the  limits  of  inte- 
gration a  and  b  ;  and  also  that  the  limits,  a  and  b,  are  both 
finite. 

A  general  investigation  of  these  exceptional  cases  will  be 
found  in  a  subsequent  chapter :  meanwhile  it  may  be  stated, 
reserving  these  exceptions,  that  whenever /(#),  i.e.  the  integral 
oif'(x)dx,  can  be  found,  the  value  of  the  definite  integral 

I  f(x)  dx  is  obtained  by  substituting  each  limit  separately 


*  In  a  subsequent  chapter  on  Definite  Integrals  a  rigorous  demonstration 
will  be  found  of  the  property  here  assumed,  namely  that  the  sum  of  these 
quantities  of  the  second  order  becomes  evanescent  in  the  limit,  and  consequently 
may  be  neglected.     Compare  also  Art.  39,  Diff.  Calc. 


32  Elementary  Forms  of  Integration, 

instead  of  x  in  /(«),  and  subtracting  the  value  for  the  lower 
limit  from  that  for  the  upper. 

A  few  easy  examples  are  added  for  illustration. 


i: 


Examples. 


x*dx.  Am.  . 

n+  i 


I    BinOdd.  „     i. 

f°      dx  ir 

J  o  a2  +  *»'  "    40 
it 

Jsw?xdx.  „     -. 

o  i                                               4 


5.  i    m&xdx. 
Jo 

6.  1 .  Bm2xdx. 
Jo 


dx 


I! 

Joi  + 


IT         I 

8  "4 

IT 
2* 


ir 

9.  cos5#ate.  ,,    — — . 

Jo  3-5 

f3    xdx  1  , 

IO-    ),t&  "  ;l0«3- 

rP  dx 

Ja\/«-a)(j8  -*) 
See  Art.  II. 

12.  I    xsin*^.  „     1. 

f  ■        ate  t 

13.  7 -,  where  a  >  b.  „ 

J  0  a  +  b  cos  6'  ya»  _  b2 

C*  dx  ir 

Jo  I  -  2a  cos#+  a1'  "     i-«2* 


Change  of  Limits.  33 

32.  Change  of  Limits. — It  should  be  observed  that  it 
is  not  necessary  that  the  increment  dx  should  be  regarded  as 
positive,  for  we  may  regard  x  as  decreasing  by  successive 
stages,  as  well  as  increasing. 

Accordingly  we  have 

["/(•)  dx  =/(«)  -/(b)  =  -  f /(*) <**.  (32) 

Jb  J  a 

That  is,  the  interchange  of  the  limits  is  equivalent  to  a  change 
of  sign  of  the  definite  integral. 
Also,  it  is  obvious  that 


re  re  rb 

<j)(x)dx=\    <p(x)dx+\    <j)(x)d%; 


and  so  on. 

Again,  if  we  assume  x  to  be  any  function  of  a  new  variable 
2,  so  that  $(x)dx  becomes  \p(z)dz,  we  obviously  have 

r>  -A.  *Z 

<j>(x)dx  -       ^{z)dz,  (33) 

J#o  J  So 

where  Z  and  z0  are  the  values  which  z  assumes  when  X  and 
x0  are  substituted  for  x,  respectively. 

dx 

For  example,  if  x  =  a  tan  *,  the  expression  — ^  be- 

(a  +  x  )s 

comes 5 — ;  and  if  the  limits  of  x  be  o  and  a.  those  of 

a?  ' 

s  are  o  and  -.     Consequently 

fa       dx  1    f7  1 

)o(a2  +  x*)t      a*  }0  a*y2 

Also,  if  we  substitute  a  -  z  for  x9  we  have 

{a  ro  ra 

<j>(x)dx  =  -      <p(a  -  *)<&  =      <j>(a  -  z)dz. 

[3] 


34  Elementary  Forms  of  Integration. 

Since  neither  x  nor  2  occurs  in  the  result,  this  equation 
may  evidently  be  written  in  the  form 

j    <p(x)dx  =  \    <p[a  -  x)dx.  (34) 

0  Jo 

For  example,  let  <j>(x)  =  sinw#,  then  <j>  I  -  -  x )  =  cosn#,  and 
we  have 

smnxdx  =     coanxdx. 
And,  in  general,  for  any  function, 

f{smx)dx  =  \f{G0sx)dx.  (35) 

Jo  Jo 

33.  Values  of    siiimx&mnxdx,  and     cosmxcomxdx. 

Since 

2  sin  mx  sin  nx  =  cos  (m  -  n)  x  -  cos  (m  +  n)  x, 
and 

2  cos  mx  cos  nx  =  cos  (m  -  n)  x  +  cos  (m  +  w)  #, 

we  have 

f  .  .         7      sin  (m  -  n)  x     sin  (m  +  5TTa? 

sm  mx  smnxdx  =  — 7 7 ; ^— , 

J  2{m  -  n)  2  (m  +  n) 

-,       f  7      sin  Cm  -  n)  x     sin  (m  +  n)  x 

and         cosmxeosnxdx  = — ) f-  +  — ^ ~ . 

I  2  (m  -  n)  2  (m  +  n) 

Hence,  when  m  and  w  are  unequal  integers,  we  have 

sin  mx  sin  nxdx  =  o,  and     cos  mx  cos  w#d#  =  o.  (36) 
"When  m  =  n,  we  have 
sin2nxdx  = 


1  -  cos  2nx    ,       x     sm  2nx 

dx  = , 

2  2  4n 

sin2 nxdx  =  -,  when  n  is  an  integer. 


Definite  Integrals.  35 

In  like  manner,  with  the  same  condition,  we  have 

cos3  nxdx  =  -.  (37) 

Again,  to  find  the  value  of 

n     

V  {x-a)  (fi-x)dx. 
Assume,  as  in  Art.  1 1,  x "•  a  cos2  0  +  j3  sin2  0  ;  then,  when 
6  =  o,  we  have  x  =  a ;  and  when  0  =  -,  x  =  |3. 

Hence,  as  in  the  article  referred  to,  we  have 
[  </(x-a)(p-x)dx -  2  O  -  a)2  j  W  0 cos20tf0. 

it  n 

Also  2  fWecosWe  =  J  ['sin'  z  6d9 


j  V(« -o)  (0  -*)&- 1  (0-a)2.  (38) 


[3  a] 


36  Examples. 


Examples. 

(i  +  cos  x)  dx  ^,11 

Am. 


J  (i  +  cos  x)  da 
(x  +  sin  x)3  2  (x  +  sin  x)2' 

2.         Izsin:r<fc.  „     Bin x  -  x cos x. 

3-       Jj~j[<to.  »     » log  (I +  *)-*. 

J.        .    .        .  .  (a  +  &r»)m+1 

(a  +  bx»)mxn-i  dx.  „    i— 1— . 

v  '  "      n(m  +  i)b 

x2dx  2  I 


f      x'dx 
J(«8+*3)3 


"  3(«3  +  ^)r 

6'       Id+^x-  "  log(tan-^ 

c           dx  .    .    \x  +  i 

7-            >  n  asin-U— --. 

f          3^*  I  ,         /**  ~  "  \ 

8'       )^  +  ^-a'  "  9l08V^T7J- 

9'       Ja2cos2s  +  ^sin2a;*  "  ^ tan_1  [a  ^nX) ' 

Jtanxdx  i  ,     ,..,, 

,     ,    .       Z     ■  ||       —77 x  loS  (« C0S    *  +  * 8m    X)' 

a  +  b  tan2a?  "  2  {b  -  a)                                   ' 

f  cos(log#)<fc  . 

J g         •  n  sin  (log  a). 


ii. 


dx 


12.  Show  that  the  integral  of  —  can  be  obtained  from  that  of  xmdx. 


"Write  the  integral  of  xmdx  in  the  form :  and,  by  the  method  of 

indeterminate  forms,  Ex.  5,  Ch.  iv.  Din7.  Calc,  it  can  easily  be  seen  that  the 
true  value  of  the  fraction  when  m  +  1  =  o  is  log  ( -  J ,  or  log  x,  omitting  the 

arbitrary  constant. 

13.  jeax  sin  mx  cos  nx  dx. 

This  is  immediately  reducible  to  the  integral  given  in  formula  (23). 

f dx  .       »  f4+5tan^ 

14.  1 : .  Ans.  -  tan-1  I  — —  /• 

J  5 +  4  sin*  3  \        3        / 


Examples. 

[x^^xdx  .       e^^'iax-  i) 

15.        ,— .  Ans.  ! '-. 

J      (i+*»)*  (1 +  a*)(i +*■»)* 

*      f    /    ,     \u  3  (« +  *)  (4*  -  3«) 

J  4.7 

f     xzdx  20  +  bx2 

I?'     J  (a +**»)*'  "     *<a+W 
Let  a  +  fo2  =  z2. 

f  (j»  +  £  cos  f)<k 

18.     1 . 

J       a  +  0  cos  x 

This  is  equivalent  to 

f  qdx     pb-  qa  C         dx 

)    b  b       J  a  +  b  cos  x' 

and  accordingly  can  be  integrated  by  Art.  18. 

f  xe* 

?    J(TT 


I  +  x 


2 


C  xdx 

J  1  +  **' 

(a  +  te»)*"  "    3a(a  +  **2)*' 

s.     f        ^  £  lo    AA+s8-  A 

J  s<v/*»  +1  3          \\/i  +»3  +  1/ 


Let    z3  +  1  =  z2. 


f    ±£L  £  1    /V1  +g»-  A 

J  x<s/x»  +  1  w        Vv/i  +  3"  +  1/ 


a  +  b  cos  0 
#  +  «  cos  0 
a  +  5  cos  0* 
The  expression  transforms  into 

dx 


24.  Integrate 
by  aid  of  the  assumption        x 


\/(a*  -  P)  (1  -  a2) 
integr 

^=  log  (x  +  Sx^T),  &c. 


accordingly,  when  a  >  b,  its  integral  is     ■  sin-1  a; ;  and  when  0  <  £,  it  is 

V^«2  -  m 


38  Examples. 


25.  Deduce  Gregory's  expansion  for  tan'1*  from  formula  (/). 
When  *  <  1,  we  have 

-  -  1  -  a*  +  s*  -  x*  +  &o.  ; 


1  +  X' 

.         C     dx  x3      x6      x* 

.:  tan-1  a;  =  I =  x + 

Ji  +  s2  3       5      7 


+  &c. 


No  constant  is  added  since  tan"1  x  vanishes  with  x. 

26.  Deduce  in  a  similar  manner  the  expansions  of  log  (1  +  a;),  and  sin-1  x. 

dd 

27.  Find  the  integral  of       : : — -. 

'  a  +  b  cos  $  +  c  sin  0 

This  can  be  reduced  to  the  form  in  Art.  18,  by  assuming  -  =  cot  a,  &c. 


f  dx 

t8.        —=.. 

{ (a  +  bx)  */  1  +  x* 


a  +  bx 
Am.  log 


</cfi  +  b2         ib-ax  +  */  (a2  +  b2){i  +**)}' 

This  can  be  integrated  either  by  the  method  of  Art.  13  or  by  that  of  Art.  23. 

29.      I  —  .  Am.  -  sec"1  ( x*  J . 

J  xy/xn  -  1  »■         \     / 

IT 

J  4"  sin  a;  ia; 
<. 
0    cos  a; 

Jo  cos  a; 
f2       dx 

33.  J0  </a*-x*d*. 

34.  I     a;  versin-1 1  -  J  dx. 

IT 

J04+ 


35- 


5  sin  a; 


3«.  \'     ».  . 

Jo  5  +  4  sin  a: 


»> 

i  l0«  2' 

H 

log  (1  +  </2)- 

>» 

I 

8* 

» 

TO* 

4  ' 

5> 

5^ 
4 

» 

il0g2. 

» 

i-©: 

(     39     ) 
CHAPTER  II. 

INTEGRATION   OF   RATIONAL   FRACTIONS. 

34.  Rational  Fractions. — A  fraction  whose  numerator 
and  denominator  are  both  rational  and  algebraic  functions  of 
a  variable  is  called  a  rational  fraction. 

Let  the  expression  in  question  be  of  the  form 

axm  +  bxm~l  +  cxm~2  +  &c. 
dxn  +  b'xn~l  +  cV*"2  +  &c.' 

in  which  m  and  n  are  positive  integers,  and  a,b, .  . .  a\  b\  . . . 
are  constants. 

In  the  first  place,  if  the  degree  of  the  numerator  be 
greater  than,  or  equal  to,  that  of  the  denominator,  by  division 
we  can  obtain  a  quotient,  together  with  a  new  fraction  in 
which  the  numerator  is  of  a  lower  degree  than  the  deno- 
minator :  the  former  part  can  be  immediately  integrated  by 
Art.  3.  The  integration  of  the  latter  part  in  general  comes 
under  the  method  of  Partial  Fractions. 

35.  Elementary  Applications. — Before  proceeding  to 
the  general  process  of  integration  of  rational  fractions,  we 
propose  to  consider  a  few  elementary  examples,  which  will 
lead  up  to,  and  indicate  in  what  the  general  method  really 
consists. 

We  commence  with  the  form  already  considered  in  Art.  7  ; 
in  which,  denoting  by  ax  and  a2  the  roots  of  the  denominator, 
the  expression  to  be  integrated  may  be  represented  by 


Assume 


(p  +  qx)dx 

(x  -a1)(x-  a2) 


p  +  qx  Ax  Ai 


(x  —  ax)  (x  -  a2)       X  —  a\      x  —  a2 


40  Integration  of  Rational  Fractions. 

Multiplying  by  (x  -  ax)  (x  -  a2)  we  get 

p  +  qx  =  -  (Aia2  +  A2ax)  +  (Al  +  A2)x. 

Hence,  we  get  for  the  determination  of  Ax  and  A2  the 
equations 

p  =  -  Aict2  -  A2ah     q  =  Ai  +  A2 ; 
whence  we  obtain 

A    -  P  +  Vai  A  P  +  4P* 

M-i  — ,         JL2  = . 

cti  —  a2  ax  —  a2 

Consequently 

C     (p  +  qx)dx     _  p  +  get!  f    jg         jt?  +  qg2  f    dx 
)(x-  ay)  (x  -  a2)        ai  -  a2  J  x  -  ax       ai  -  a2  J  x  -  a2 

=  ~ -\{P  +  Oa1)log(x-al)-~{p  +  qa2)log(x-a2)\. 

«1  —   «2    \  J 


In  like  manner 


p  +  qx2  Ay  Ax 


(x2  -  ai)  (x2  -  a2)      x2  -  ai      x2  -  a2 
where  Ax  and  «4a  have  the  same  values  as  above ;  hence 

f(p  +  qx2)  dx       _  p  +  qax  C     dx         p  +  qa2  f     dx 
(x2  -  ai)   (x2  -  a2)       at  -  a2)  x2  -  ax       ai  -  a2]  x2  -  a2 

But  each  of  the  latter  integrals  is  of  one  or  other  of  the 
fundamental  forms  (/)  and  (h)  of  Chapter  I. ;  hence  the 
proposed  expression  can  be  always  integrated. 

Again,  let  it  be  proposed  to  integrate  an  expression  of 
the  form 

( p  +  qx  +  rx2)  dx 


(x  -  ai)(x  -  a2)(x-  a3)' 
We  assume 

p  +  qx  +  rx2  Ax  A2  A3 


(x  -  ax)  (x  -  a2)  (x  -  a3)       X  -  ai      X  -  a2      x-a* 


Elementary  Applications.  41 

then  clearing  from  fractions,  and  identifying  both  sides  by 
equating  the  coefficients  of  x2,  of  #,  and  the  part  independent 
of  x,  at  both  sides,  we  obtain  three  equations  of  the  first 
degree  in  Alt  A2i  A3,  which  can  be  readily  solved  by  ordinary 
algebra  ;  thus  determining  the  values  of  Ax,  A2>  A3  in  terms 
of  the  given  constants. 
By  this  means  we  get 

f       ( p  +  qx  +  rx2)  dx  .  f    dx         .  f    dx  .  f    dx 

\- K-£~ ry r  =  AA +  A2\ +  A3\ 

J  {x  -  ai)  [x  -  a2)  {x  -  a3)  J  x  -  en  J  X  -  a2  J  x  -  a3 

=  Ax  log  (x  -  en)  +  A2  log  (x  -  a2)  +  Az  log  (*  -  a3). 

We  shall  illustrate  these  results  by  a  few  simple  examples. 
Examples. 

••  fe^n&i"       ^.5-iog(i-3)+iiog(,+J). 

C       xdx  "5  i 


#  —  I       I 


Ji^r*  w  ;l0g^TT--2tan"Ia;- 

J<?#  r         ,        i  x 

— t— ^— — .  „     -  tan-1^  -  -  tan-1  -. 

f    xdx  I         a;2  -  1 

J^T7-  »   Jlos 


5- 


6         a^-2)dx 
)  x4-  -  3*2  -  4 


4     °  *2  +  I 

I,       /z2-2^ 


+  tan-1  x. 


1.     /*3-2\ 

-  log  [  — ] 

f(z3  +  X-l)dX  r  I,        ,  ,       T,  ■       . 

Here  the  denominator  is  equal  to  #(«  —  2)  (**+  3)  ;  and  we  have 
x~  +  x  -  1        _  -^1        -^2  -^3 


*(*  -  a)  (*  +  3)       *       x  -  2      #  +  3 ' 


42  Integration  of  Rational  Fractions. 

hence         xl  +  x  -  I  =  A\{&  +  x  -  6)  +  A2x(x  +  3)  +  ^3^ (*  -  2)  ; 
.*.  the  equations  for  determining  Ax,  A2  and  A%  are 

Ai  +  A2  +  A3=  I,        -4i  +  3^2  -  2^3  =  fi         6  Jj  =  1, 
whence  we  get 

Ax  =  -,  ^2  =  -,  A3  =  -. 

023 

0         f  {2X3  +  2x*  +  a,x  +  1)  dx  .        m     .      .  . 

8.  v 5 2 2 — .        A.m.  x2  +  log  (s2  +  x  +  1). 

J  x'  +  X  -^  1 

We  now  proceed  to  the  consideration  of  the  general 
method,  and,  as  it  is  based  on  the  decomposition  of  partial 
fractions,  we  begin  with  the  latter  process. 

36.  Partial  Fractions. — The  method  of  decomposition 
of  a  fraction  into  its  partial  fractions  is  usually  given  in 
treatises  on  Algebra ;  as,  however,  the  process  is  intimately 
connected  with  the  integration  of  a  large  class  of  expressions, 
a  short  space  is  devoted  to  its  consideration  here. 

For  brevity,  we  shall  denote  the  fraction  under  con- 
sideration by  -A-4. 

Let  aiy  o2,  a3,  .  .  .  a»  denote  the  roots  of  <p(x) ;  then 

<t>(x)  =  (x  -  ax)(x  -  a2)(x  -  a»)  .  .  .  (x  -  a»).  (1) 

There  are  four  cases  to  be  considered,  according  as  we 
have  roots,  (1)  real  and  unequal;  (2)  real  and  equal;  (3) 
imaginary  and  unequal ;  (4)   imaginary  and  equal. 

We  proceed  to  discuss  each  class  separately 

37.  Real  and  Unequal  Roots. — In  this  case  we  may 
assume 

P(n„\  A  A  A  A 

(*) 

where  AXi  A2i  .  .  .  .  An  are  independent  of  x.  For,  if  the 
equation  be  cleared  from  fractions  by  multiplying  by  #(#), 
on  equating  the  coefficients  of  like  powers  of  x  on  both 
sides  we  obtain  n  equations  for  the  determination  of  the  n 
constants  Alf  A2)  .  .  .  An. 


f(x)       Ax         A2          A3 

An 

— — -  = 1- h h  .  , 

(j>  (x)      x  —  ax      x  -  a2      X  -  a3 

X-  an 

Real  and  Unequal  Boots.  43 

Moreover,  since  these  equations  contain  Ah  A2,  &c,  only 
in  the  first  degree,  they  can  always  be  solved  :  however,  since 
the  equations  are  often  too  complicated  for  ready  solution, 
the  following  method  is  usually  more  expeditious  : — 

The  question  (2),  when  cleared  from  fractions,  gives 

f(x)  =  Ax(x-  a2) (x -  a3)  .  .  {x -  a„)  +  A2(x- ai) (x  -  a3)  .  .  (x  -  a„) 
+  &c.  +  An  (x  -  ai)  (x  -  a2)  .  .  (x  -  an.i)  ; 

and  since,  by  hypothesis,  both  sides  of  this  equation  are 
identical  for  all  values  of  x}  we  may  substitute  ax  for  x 
throughout;  this  gives 

/(ai)  =  Ai(ai  -  a2)(a1  -  a3)  .  .  .  (ai  -  an), 
In  like  manner,  we  have 

A     _  fM  A     _/M  A     _  /(«»)  /_. 

0  (a2)  tf>  (a3)  0  (a„) 

Hence,  when  all  the  roots  are  unequal,  we  have 
Accordingly,  in  this  case 

firl  *  -  tH  log  (•  -  «>)  +  f-¥k log  (*  -  «•)  +  4c 
J0O»)       #'(0,)    6V       '    4>'(a2)  s^      ' 

+  -g^  log  (•-«,).  (5) 

The  preceding  investigation  shows  that  to  any  root  (a), 
which  is  not  a  multiple  root,  corresponds  a  single  term  in  the 
integral,  viz. 

£g  log  (*-«); 


•(4) 


44  Integration  of  Rational  Fractions. 

one  which  can  always  be  found,  whether  the  remaining  roots 
are  known  or  not ;  and  whether  they  are  real  or  imaginary. 

38.  Case  where  Numerator  is  of  higher  Degree 
than  Denominator. — It  should  also  be  observed  that  even 
when  the  degree  of  x  in  the  numerator  is  greater  than,  or 
equal  to,  that  in  the  denominator,  the  partial  fraction  cor- 
responding to  any  root  (a)  in  the  denominator  is  still  of  the 
form  found  above. 

For  let 

m  H  Wf 

where  Q  and  R  denote  the  quotient  and  remainder,  and  let 

A  R 

be  the  partial  fraction  of  — j-r  corresponding  to  a  single 

root  a ;  then,  by  multiplying  by  <j>(x)  and  substituting  a  in- 
stead of  x,  it  is  easily  seen,  as  before,  that  we  get 

For,  example,  let  it  be  proposed  to  integrate  the  ex- 
pression 

x5dx 


X3  -  2X2  -  $x  +  6 
Here  the  factors  of  the  denominator  are  easily  seen  to  be 

x  -  1,     x  +  2,  and  x  -  3  ; 
accordingly,  we  may  assume 

x5  2  Q       A  B  C 

=  or  +  ax  +  p  + + + 


x3  -  2x2  -  $x  +  6  a?  —  1      x  +  2      x  -  3 

To  find  a  and  /3,  we  equate  the  coefficients  of  xi  and  x3  to 
zero,  after  clearing  from  fractions :  this  gives,  immediately, 
a  =  2,  and  )3  =  9. 

Again,  since  <j>(x)  =  x3  -  2x%  -50  +  6,  we  have 

<t>\x)  =  3X2  -  4x-  5. 


Real  and  Unequal  Boots.  45 

Accordingly,  substituting  1,-2,  and  3,  successively  for  x 
in  the  fraction 

x* 
3#2  -40-5' 


1    *--3£.    c  =  243 


we  get 

and  hence 

x5  .  1  32  243 

7  =  Of +205  +  9  -77 - ; r  +  y rj 

x*  -  2xz  -  $x  +  6  6(a?-i)      15  (a; +2)      10(05-3) 

f  x5dx  x3      2  log  (a;-  1) 

"  *  J  x3  -  2X*  -  50?  +  6      3  y  6 

32  243 

-—log  (05  +  2) +—log(oj-3). 

39.  Case  of  Even  Powers. — If  the  numerator  and 
denominator  contain  x  in  even  powers  only,  the  process  can 
generally  be  simplified ;  for,  on  substituting  %  for  o?2,  the 
fraction  becomes  of  the  form 

m 

Accordingly,  whenever  the  roots  of  0(2)  are  real  and 
unequal,  the  fraction  can  be  decomposed  into  partial  fractions, 
and  to  any  root  (a)  corresponds  a  fraction  of  the  form 

m  r 

$'(a)  s  -  «' 
The  corresponding  term  in  the  integral  of 


is  obviously  represented  by 

/(«)    f    dx 
${*)    \x2-a 


46  Integration  of  Rational  Fractions. 

This  is  of  the  form  (/)  or  (h),  according  as  a  is  a  positive 
or  negative  root. 

The  case  of  imaginary  roots  in  <f>(z)  will  be  considered  in 
a  subsequent  part  of  the  chapter. 

It  may  be  observed  that  the  integrals  treated  of  in  Art.  5 
are  simple  cases  of  the  method  of  partial  fractions  discussed 
in  this  Article. 


Examples. 
f  {2x  +  i)dx 

J  X*  +  Z2  -  2X 

Here  the  factors  of  the  denominator  evidently  are  x}  z  -  1,  and  x  +  2 ;  we 
accordingly  assume 

_2£_fj__  A         B  G 

Zz  +  X2  -  IX        X        X  —  I        x  +  2 

Again,  as  <£  (x)  =  x*  +  x2  -  zx,  we  have  <p'(x)  =  3a;2  +  iz  -  2  ; 
•    /(*)  =       **  +  3 

<t>'(z)        $X*  +  2T  -  2 

Hence,  hy  (3)  we  have 

2  3  6 

consequently 

IVix  +  3)  dx  3 ,  5 ,      ,  1  ,      /          » 

*  +  +  -2M—\**  *   +  J'08  (—  *>"  8  l0S  (X  +  * 

f  dx 

2'    '  J  {z*  +  a*){z*  +  b*)' 

Here 

-     '      /    ' !_V 

(z2  +  IFjfoP  +  £2)      a2  -  b*  W  +  P      «*  +  «V  ' 
hence  the  value  of  the  required  integral  is 

f=u6,,*i(t) -?*■*©!■ 


Multiple  Real  Roots.  47 

Substitute  z  for  x2  and  the  transformed  integral  is 


J2(2  + 


dz 


a)  {z  +  b) 
Consequently  the  value  of  the  required  integral  is 

.2 


4- 

f  (3Z2  -  1)  dx 

J  x2  -  3x  +  2' 

f  (#2  —  3)  ^ 

5' 

J  x3  -  7#  +  6* 

6. 

j"      (22;  +  1)  dx 

J  #(#  +  i)(#  +  : 

7- 

(•        £2*fo; 

Js*-a;2-  12* 

8. 

Let 

_J .     /s2  +  s\ 

2    («  -  *)        g     U2   +   «/  ' 

^WS.  3«?  +  II  log  (X  -  2)  -  2  log(#  -  I). 


„     \  log  (ar-  I)  +  -log(*  -  2)  +  -^-log(*  +  3). 


„     -  log  x  +  log  («  +  1)  --log  (x  4  2). 
2  2 


f   dx(a  + 


r 


2 


40.  Multiple  Real  Roots. — Suppose  $(%)  has  r  roots 
each  equal  to  a,  then  the  fraction  can  be  written  in  the  shape 

m 


(x  -  a)r\p{x) 
In  this  case  we  may  assume 


fix)  mx   ;  , \M%  '  .         ,   f r        P 


(x  -  a)r^  (x)       (x  -  a)r       (x  -  a)'-1      '  '  '      x  -  a      1//  (#)' 

where  the  last  term  arises  from  the  remaining  roots. 

For,  when  the  expression  is  cleared  from  fractions,  it  is 
readily  seen  that,  on  equating  the  coefficients  of  like  powers 
at  both  sides,  we  have  as  many  equations  as  there  are 
unknown  quantities,  and  accordingly  the  assumption  is  a 
legitimate  one. 


48  Integration  of  Rational  Fractions. 

In  order  to  determine  the  coefficients,  Mif  M2,  &c.  .  .  .  Mn 
clear  from  fractions,  and  we  get 

f(x)  =  M4{x)  +M2(x  -  a)\P(x)  +  M3(x-ay^(x)  +  &o. . . .      (6) 

This  gives,  when  a  is  substituted  for  x, 

f(a)=M4{a),0TMlJ^L.  (7) 

Next,  differentiate  with  respect  to  x,  and  substitute  a 
instead  of  #  in  the  resulting  equation,  and  we  get 

f{a)  =  M4\a)  +  M4(a)  ;  (8) 

which  determines  M2. 

By  a  second  differentiation,  Mz  can  be  determined ;  and 
so  on. 

It  can  be  readily  seen,  that  the  series  of  equations  thus 
arrived  at  may  be  written  as  follows — 

f(a)=M4'(a)    +l.M4(a), 
/'(a)  =  Mrf'ia)   +  2  .  M4\a)  +  I  .  2.M4{a\ 
/"'(«)  =  M4'"(a)  +  3  •  M4"(a)  +  2.3  .  Mrf  (a)  +1.2. 3  .  Jf^(a), 
/*(a)  =  M.f^a)  +  4.M4'"(a)  +  3-4-M4"W  +  2.3.4.Ifrf'(«i) 

+  1.2.3.4.  M4{a), 

in  which  the  law  of  formation  is  obvious,  and  the  coefficients 
can  be  obtained  in  succession. 

The  corresponding  part  of  the  integral  of 

f(x)  dx 


(x  -  a)r\p  {x) 
evidently  is 

Mr.,        I     Mr-2  Mx 

if, log  (»-,)-  — --  —  -•■•-(,,_  l)(g_ar.    (9) 

If  <j>(x)  have  a  second  set  of  multiple  roots,  the  cor- 
responding terms  in  the  integral  can  be  obtained  in  like 
manner. 


Imaginary  Roots.  49 

41.  Imaginary  Roots. — The  results  arrived  at  in 
Art.  37  apply  to  the  case  of  imaginary,  as  well  as  to  real 
roots ;  however,  as  the  corresponding  partial  fractions  appear 
in  this  case  under  an  imaginary  form,  it  is  desirable  to  show 
that  conjugate  imaginaries  give  rise  to  groups  in  which  the 
coefficients  are  all  real. 

Suppose  a  +  b  */-  1  and  a  -  b  >/-  1  to  be  a  pair  of  con- 
jugate roots  in  the  equation  <p(x)  =  o ;  then  the  corresponding 
quadratic  factor  is 

(x  -  a)2  +  b2 ;  which  may  be  written  in  the  form  x2  +  px  +  q. 

We  accordingly  assume 

(j>(x)  m  (x2  +px  +  q)\p(x)f 
and  hence 

f(x)        Lx  +  M       P 
$(x)      x2  +  px  +  q      Q' 

P 

where  -^  represents  the  portion  arising  from  the  remaining 
H 

roots,  and  — is  the   part   arising   from  the   roots 

x2  +  px  +  q  s  ° 

a  ±b  */-  1 . 

Multiplying  by  $(x)  we  get 

p 

f{x)  =  (Lx  +  M)  \p  (x)  +  (x2  +  px  +  q)  -^  ^  (x).         (10) 

If  in  this,  -  {px  +  q)  be  substituted  for  x2,  the  last  term 
disappears ;  and  by  repeating  the  same  substitution  in  the 
equation 

f(x)=^{x){Lx  +  M), 

it  ultimately  reduces  to  a  simple  equation  in  x :  on  identify- 
ing both  sides  of  this  equation,  we  can  determine  the  values 
of  L  and  M. 

42.  In  many  cases  we  can  determine  the  coefficients  Z,  M 
more  expeditiously,  either  by  equating  coefficients  directly, 
or  else  by  determining  the  other  partial  fractions  first,  and 
subtracting  their  sum  from  the  given  fraction. 

It  will  also  be  found  that  the  determination  of  many 
[4] 


50  Integration  of  Rational  Fractions. 

integrals  of  this  class  can  be  much  simplified  by  a  trans- 
formation to  a  new  variable,  or  by  some  other  suitable 
expedient. 

Some  elementary  examples  are  added  for  the  purpose 
of  illustration. 


Examples. 

f  xdx 

U  J  (i+*)(i  +  *V 

Assume 

x  A        Lx  *  M 


(i  +x){i  +  a;2)  _  i-i  x       I  4  x1' 
clearing  from  fractions,  this  becomes 

x=  A  (i  +  x2)  +  {Lx  +  M)(i  +  x). 
Equate  the  coefficients,  and  we  get 

L  +  A  =  o,        L  +  M=i,        A  +  M=o. 
Hence 


L-1-,      *-i,      Am- 1 

2*  2*  2 


and  accordingly 


x  II  i  i +x 

(l  +  x)(l  +  X2)  ~~       2  l+X        2I+»2' 

f  xdx  i        I   i  +•  #2  I      I 

f    dx 


2. 


Let 

i  ^  Lx  +  M 


i  +x3      i+x      i  —  #  t  x2 ' 

consequently,  ^4  =  -,  by  formula  (3).     Substituting  and  clearing  from  fractions 
we  have 

3  =  1  -  x  +  x2  +  3  {Lx  +  M){i  +  x) ; 
hence,  dividing  by  1  +  #,  we  have 

2  -  3;  =  3  {Lx  +  M). 


Imaginary  Boots.  51 

Consequently 

idx    _  i  j*  dx        i  {'(2  -x)dx 
i  +  a;3  ~~  3  J  I  +  a;      3  J  i  -  a;  +  a;2 

=  -log(i  +  *)-]Uog(i  -x  +  x2)  +  -^ztan-1(^-^-). 
3  b  /3  V  Vl  ' 

C     dx                           .        1.       /  1  +x  +  x*\  1  /2*  +  l\ 

3.       I .  Ans.  -  log    3    +  — -  tan"1    — j=  J . 

This  can  be  got  from  the  last  by  changing  the  sign  of  x. 

f     dx 

4-  J  T=& 

In  this  case  we  have 

1      _  1  /     1  1     \ 

1  —  *•  ™ 2  \i  —  **     1  +  *y ' 

f  aj'dir  .        I  .      (    (*4-i)2    )  1  (2*4+1) 

c.         — .  Ans. —log] —L—  \  + — — itan-1] —  L 

5         J*12-i  24    6Ub  +  *4  +  iJ      4v/3  I  v/ 3  ) 

Let  a;4  =  2j  and  the  integral  becomes 

1  f    z<fe 

C  x2dx 

J(#-i)*(«»  +  iy 

Assume 

a:8  .4  Z        Xs+Jf 

+ 


(*-  i)2(x2  +  I)      (<c-l)2      a?- 1        i  +  a;2* 

To  find  L  and  Jf,  clear  from  fractions,  and  by  Art.  41  the  values  of  L  and  M 
are  found  by  making  xz  =  -  1  in  the  following  equation : 

a;2  =  {Lx  +  JK)(*  -  I)2. 

This  gives  immediately  Z  =  — ,     M  =  o. 
2 

Again,  by  Art.  40,  we  get  immediately  A  =  -. 


To  find  J?,  make  x  =  o  in  both  sides  of  our  identity,  and  we  get 

M;    .'. 
[4a] 


o  =  A-  £  +  M;    .:B  =  A  =  -. 
2 


52  Integration  of  Rational  Fractions. 

Finally 


II  ii  IX 

+  - 


(*  -  l)2(*2  +1)        2  U  -  02        2  *  -  I        2    I  +  X*  * 

•••I(Tr^gr7o--i.-ri+ik*<'-')-?log(**+,)- 

f  dx 

J  a8  +  a;7  -  a:4  -  a;3* 

Here  the  denominator  is  easily  seen  to  be  a?(x-  i)(s  +  i)2(a:2  +  i),  and  the 
expression  becomes 

f  dx 

)x*(x-  i)(*+  i)2(*2  +  i)* 

Assume  a;  =  - ,  and  the  transformed  expression  is  evidently 

z 

f Z«<fe 

Iff- I)(« +!)»(«»+  I)* 

The  quotient  is  easily  seen  to  be  z  -  i ;  and,  by  the  method  of  Art.  38,  we  may 


(z  -  1  )(z  +  1  )2  (z2  +  1)  "  Z  "  l  +  z  -  1  +  (z  +  i)2  +  z  +  1  +    z«  +  I  * 
Hence  (Arts.  37,  40),  we  have 

1         ,>         1 

Next,  L  and  Jf  are  found  by  making  z2  =  -  1,  in  the  equation 

z«  =  (Xz  +  jf)(z-  »)(*  + J)2; 

.-.  1  =  2(Zz  +  M)(z  +  1)  =  2  {Z*2  +  (Z  +  Jf)z  +  Jf }, 
which  gives 

i+Jf=o,     Z-Jf=--; 

4  4 

In  order  to  find  the  remaining  coefficient  C,  we  make  z  =  o,  when  we  get 

o  =  -i-A  +  £+C+M;        .V0«|. 


Multiple  Imaginary  Boots.  53 

hence  we  have 

z6  i  I  9  s  -  I 

-.Z-I+— r-      ,      ,    .„  + 


(*-l)(«.+ !)«(«»+ 1)  '8(Z-i)      4(^  +  i)2      8(z+i)      4(z2+i)' 

f  zsrfz  Z2  I  I 


+  |  log  (f  +  i)  -  i  log  («»  +  I)  +  -  tan-1; 

OO4 


Hence 

dx 


f dx  1        1  a;  1         1  -  a=2  a;  +  1      X .    ■    1 

J  ajS  +  ^-^-a3  "  2as*  ~  x  +  4(x+  I)  +  8   °g  i  +  a*  +  °S~x~  +  4   ^  * 

8.  — - — — -i ,.  -4«s.  -  log . 

J  (#  -  i)2  (a?  +  3)  2     e  a;  +  3      x  -  I 


43.  Multiple    Imaginary   Roots. — To   complete   the 

fix) 
discussion  of  the  decomposition  of  the  fraction  —j-t,  suppose 

the  denominator  <p(x)  to  contain  r  pairs  of  equal  and  imaginary 
roots,  i.  e.  let  the  denominator  contain  a  factor  of  the  form 
\{x  -  a)2  +  b2}r;  and  suppose  <j>(x)  =  {{x  -  a)2  +  b2}r  $x(x) 
In  this  case  we  assume 

f(x)  Lxx  +  Ml  L2x  +  M2 


(x  -  a)2  +  b*}rfr(x)       {(x-a)2  +  b2}r      {{x  -  a)2  +  b2}*-1 


Lrx  +  Mr  P 

+   •    .    •    +    5 r= ™   + 


.(x~a)2+b2     0i(») 

the  remaining  partial  fractions  being  obtained  from  the  other 
roots. 

There  is  no  difficulty  in  seeing  that  we  shall  still  have 
as  many  equations  as  unknown  quantities,  Xx,  Mi,  L2,  M2) . . . 
when  the  coefficients  of  like  powers  of  x  are  equated  on  both 


To  determine  Lif  Ml9  L2i  &c. ;  let  the  factor  (x  -  a)2  +  ¥ 
be  represented  by  X,  and  multiply  up  by  Xr,  when  we  get 

^r  =  LiX  +  Mi+  [L2x  +  M2)X  +  &c.  +  (£r  x  +  Mr)  X^  +  ^- •     ( I  i) 


54  Integration  of  Rational  Fractions. 

The  coefficients  Lt  and  Mx  are  determined  as  in  Art.  4 1 . 
To  find  L2  and  M2 ;  differentiate  with  respect  to  x,  and  sub- 
stitute a  +  by/  -  1  for  x  in  the  result,  when  it  becomes 


d_ 
dx 


r  f(x\  "I 
'— j-r    =  Zx  +  2(ar0  -  a)(L2x0  +  M2), 

l<pi{X)Jo 


where  #0  =  a  +  Jy^  -  1 . 

Hence,  equating  real  and  imaginary  parts,  we  get  two 
equations  for  the  determination  of  L2  and  M2.  By  a  second 
differentiation,  L3  and  Mz  can  be  determined,  and  so  on. 

It  is  unnecessary  to  go  into  further  detail,  as  sufficient  has 
been  stated  to  show  that  the  decomposition  into  partial  frac- 
tions is  possible  in  all  cases,  when  the  roots  of  <p(x)  =0  are 
known. 

The  practical  application  is  often  simplified  by  transfor- 
mation to  a  new  variable. 

44.  The  preceding  investigation  shows  that  the  integra- 
tion of  rational  fractions  is  in  all  cases  reducible  to  that  of 
one  or  more  fractions  of  the  following  forms: 

dx  dx  (A  +  B)dx         (Lx  +  M)dx 


x-J     (x-af     {x-ay  +  b2'     {(x  -  a)2  +  b2}r' 

The  methods  of  integrating  the  first  three  forms  have  been 
given  already.  We  proceed  to  show  the  mode  of  dealing 
with  the  last. 

45.  In  the  first  place  it  can  be  divided  into  two  others, 

L  [x  -  a)dx  (La  +  M)dx 

{{x-af  +  Vy*  [(x-a*)+by 

The  integral  of  the  first  part  is  evidently 


2(r_  1)  [(x  -  a)2  +  b2}r~l 


To  determine  the  integral  of  the  other  part,  we  substitute 
2  for  x  -  a,  and,  omitting  the  constant  coefficient,  it  becomes 

f      dz 

J  (z2  +  by 


Multiple  Imaginary  Roots. 


55 


Again 


dz 


(z2  +  b2)r     b2 


(z2  +  b2-z2) 


(*2  +  b*y 

But  we  get  by  integration  by  parts 

f     z2dz  zdz  i 

J  (s2 + vy  =  J z '  (s2  +  b2y  = "  ^(T- 1) j 


i  r     c?s        i 
=  &\](s2  +  fry-1'?? 


z2dz 


(s2  +  £2)'- 


zd 


(s2  +  fc2)'-1 


2(r-  i)(z2  +  b'2)r-1      2(r-i), 
Substituting  in  the  preceding,  we  obtain 
ds  2r- 


(z2  +  b2y~i' 


{z2  +  b2)r      2(r 


-3     f       dz 
i)b>J{zi  +  bi)r-1 


2(r-i)b'i(z2  +  by-1' 


(12) 


This  formula  reduces  the  integral  to  another  of  the  same 
shape,  in  which  the  exponent  r  is  replaced  by  r  -  i.  By 
successive  repetitions  of  this  formula  the  integral  can  be  re- 
duced to  depend  on  that  of 


(z2  +  b2) 

The  preceding  is  a  case  of  the  method  of  integration  by 
successive  reduction,  referred  to  in  Art.  19.  Other  examples 
of  this  method  will  be  found  in  the  next  Chapter. 

The  preceding  integral  can  often  be  found  more  expedi- 
tiously by  the  following  transformation  : — Substitute  b  tan  9 

dz 
for  2,  and  the  expression  j-% 7—  becomes,  obviously, 

The  discussion  of  this  class  of  integrals  will  be  found  in 
the  next  Chapter. 

46.  We  shall  next  return  to  the  integration  of     v  /     , 

#(# ) 
which  has  been  already  considered  in  Art.  39  in  the   case 


56  Integration  of  Rational  Fractions. 

where  the  roots  of  <j*(z)  are  real.     To  a  pair  of  imaginary 
roots,  a  ±  b  */-  i ,  corresponds  a  partial  fraction  of  the  form 

(Ax2±B)dx  (Ax2  +  B)dx 

(x2  -  a)2  +  P>     °r  x*  -  2ax2Tc2' 

where  c*  =  a2  +  b2. 

In  order  to  integrate  this,  we  assume  a  =  c  cos  20,  when 
the  fraction  becomes 

(Ax2  +  B)  dx 

X4,  -  2X2C  COS  20  +  C2' 

The  quadratic  factors  of  the  denominator  are  easily  seen 
to  be 

x2  -  2xV c  cos  0  +  c,  and  x2  +  ix  \/c  cos  <f>  +  c. 
Accordingly  we  assume 
Ax2  +  B  Lx  +  3f  L'x  +  M' 


X4,  -  2X2C  COS  20  +  C2      X2  -  2X  ^/c  COS  0  +  C      X2  +  2X ^/ 0  COS  0  +C 

hence  it  can  be  seen  without  difficulty  that 

4  C-  COS  0  2C 

and  after  a  few  easy  transformations,  we  find 

f       (^#2  +  B)dx  Ac  -  B  1      /V  -  2a;  >v/^  cos  0  +  c 

J  #*-2a;2ccos20  +  c2      8cos0c*     S\a*  +  2x </c  gob  <j>  +  c, 

4sm0c*  V      c-a? 

47.  Integration  of     {x  _  a)t\x  _  bf 

This  expression  can  be  easily  transformed  into  a  shape 


dx 

Integration  of -. r— -: —  .  57 

*  J  (x  -  a)m  (x  -  b)n 

which  is  immediately  integrable,  by  the  following  substitu- 
tion : — 

Assume  x  -  a  =  (x-  b)  z;  then 

a  -  bz  (a-  b)z  7     a  -  b     7       (a-b)dz 

x=  ;  .'.  x-a  =  - — ,   x-b  = ,   ax  =  ~, rr-; 

I  -*  i  -  z  i  -z  (i  -  zy 

and  the  expression  transforms  into 

(i  -  z)"""^ 
(a  -  5)m+w-12m# 

Expand  the  numerator  by  the  Binomial  Theorem,  and  the 
integral  can  be  immediately  obtained.     (Compare  Art.  4.) 
For  example,  take  the  integral 

dx 
(x  -  a,y(x-  by 

Here  the  transformed  expression  is 

f(i  -  z)zdz 


{a-byz*' 


or 


^rjjij (?  ~ I +  3 "*)* "  (^V  I?"  3* +  3  logs  +  i 

/*» /» 

Substituting =   for  2,  the  integral  can  be  expressed  in 

x     0 

terms  of  x. 

48.  Integration  of     - 


(a  +  cx2)n 

where  m  and  n  are  integers. 

Let  a  +  ex2  =  z,  and  the  expression  becomes 

(z-a)mdz% 

2Cm+lZn      ' 

a  form  which  is  immediately  integrable  by  aid  of  the  Bino- 
mial Theorem. 


58  Integration  of  Rational  Fractions. 

It  is  evident  that  the  expression  is  made  integrable  by  the 
same  transformation  when  n  is  either  a  fractional  or  a  nega- 
tive index. 

It  may  be  also  observed  that  the  more  general  expression 
f(x2)  xdx 

(a  +  ex' 

f{x2)  denotes  an  integral  algebraic  function  of  x2. 


can  be  integrated  by  the  same  transformation,  where 


Examples. 

Jx^dx                                         .               a*  x2        „,      ,  . 

tt 5i-  Ans.  — +  -  +  a2  log  (a2  -  x2). 

(a2-x-)-  2(a?-x-)      2 

fidx  la 


f     x*dx 
J  (a  +  ex2)*' 


(a  +  ex2)*  "        4c2  (a  +  ex2)2      6c2  (a  +  ex2)3 

f     x5dx  1  11 

dx 


49.  Integration  of 


af  -  i' 


where  n  is  a  positive  integer. 

Suppose  a  an  imaginary  root  of  xn  -  1  =  o,  then  it  is  evi- 
dent that  a"1  is  the  conjugate  root :  also,  by  (3),  the  partial 
fraction  corresponding  to  the  root  a  is 


a 

or 


nan~l(x  -  a)'  n(x  -  a)' 

If  to  this  the  fraction  arising  from  the  root  a"1  be  added, 
we  get 

I  (     a  a~l     )  I  {      x(a  +  a"1)  -  2 

-  { + ,},  or  -    T/ —     t\ 

n  [x  -  a      x  -  a"1)  n  [x2  -  (a  +  a  *)  X  +  1 

But,  by  the  theory  of  equations,  a  is  of  the  form 

2&7T  / .     2kir 

cos +  v-  1  sin , 

n  n 


dx 
Integration  of— .  59 


where  k  is  any  integer ; 


ikn 
.'.   a  +  aT1  =  2  COS  — . 

n 

Hence,  if  9  be  substituted  for  — ,  the  preceding  fraction 

n 

becomes 

2         x  cos  9  -  I 


The  integral  of  this,  by  Art.  7,  is 

cos  9 ,  „      .„     2  sin  9  ,      Jx  -  cos  9s 

n 


i      ,  a      ..x     2  sm  0  ,     j  /*  -  cos  0\ 

log  (1  -  2#  cos  0  +  a?-) tan  *   — : — ^ —  . 

0  '         n  \    BW.V    J 


There  are  two  cases  to  be  considered,  according  as  n  is 
even  or  odd. 

(1).  Let  n  =  ir  :  in  this  case  the  equation  x%r  -  1  =  o  has 
two  real  roots,  viz.,  +  1  and  -  1  ;  and  it  is  easily  seen  that 

[     dx  1   .      x  -  1        I   _        kir,      ,  kn        lN 

-=- =  —  log +  —  S  cos  —  log  (I  - 2X  cos  —  +  X*) 

J  aF  - 1      2r     °  x  +  1      2r  r      ° K  r         ' 

[  k^ 

,  /  x  -  cos  — 

--Ssin^tan-    *JL\  (13) 

r  r  \       .    kir 

\  sinv 

where  the  summation  represented  by  S  extends  to  all  integer 
values  of  k  from  i  tor-  i. 

(2).  Let  n  =  2r  +  i,  we  obtain 


f      tf#         log(aj-i)         I      „          2krr  ,      /                   2&7T       A 
3=1 =  -^ L+  SCOS log    I-2#C0S +  X*) 

J#2m-i        2r+i        2r+i  2r+i    &\  2r+i       J 


2r  +  1 


60  Integration  of  Rational  Fractions. 

where  the  summation  represented  by  2  extends  to  all  integer 
values  of  k  from  i  up  to  r. 

50.  Integration  of ,  where  m  Is  less  than  n  +  1. 

xn  -  1 

As  before,  let  a  be  a  root,  and  the  corresponding  partial 

am~l  am 

fraction  is     m  ,  . r  or  — r  ;  hence  the  partial  fraction 

no*1'1  (x-a)      n(x  -  a)  r 

arising  from  the  conjugate  roots,  a  and  a"*,  is 

1/   am  cl™    \  _  I    x{am  +  am)  -  (a"1-1  +  glm'1)) 

n\x  -  a      x-a~l)      n  x2  -  (a  +  a-1) x  +  1 

2  x cos mO  -  cos (m  -  i)0 
«        x1  -  2X  cos  6/  +  1       ' 

where  0  is  of  the  same  form  as  before. 

The  corresponding  term  in  the  proposed  integral  is  easily- 
seen,  by  Art.  7,  to  be 

j  l  /p COS  \j\ 

-  Jcosm01og(#2-  2#cos0+  1)  -  2  sinmfltan-1 — =-"5-[*      (J5) 

n 
By  giving  to  k  all  values  from  1  to  —  1,  when  n  is  even,  and 

M     T 

from  1  to  when  n  is  odd,  the  integral  required  can  be 

written  down  as  in  the  preceding  Article. 


Examples.  61 

Examples. 

f       dx  a      li     (x  +  2\ 

*«        1  -s — p t.-        Am. -log  [ J. 

J  x2  +  6x  +  8  2     &  \#  +  4/ 

2*       j^,^.^  »     alog(*-a)  +  log(*  +  i). 

f(^  +  £a;2)^  -4,  Ba-Ab,      , 

J    x(a  +  &£■*)  «  2«o 

Ix2dx                         I,     a;-l      a/2,       ,  /    #    \ 
-7 ^ .  „     -log +  - — tan-1!——:). 

x*  +  x2-2  "     6    6x  +  i         3  Vv/z/ 

f<for  i      .      #s  4  2:^/2  +  1         I  ,  fx\/ 2\ 

*    +  1  V 2        x2-xy/2  +  I     2/2  \l-x°~J 

f    (2x-5)dx  7         ,  IT  w  /g+M 

'       J(*43)(*+i)3'       "     2(^+i)+4      gU+3/' 

f         <fo  1  1  /     x2     \ 

7*        )x(a+  bx2)2'  "      2a  (a  +  bx2)  +  za?  °S  \a  +  bx2)  ' 

8         f         dx  I  .      ( _J^__\  1 

.      •       (z-by^dz 
Let  0  4-  fon  =  #%  and  the  transformed  expression  is  -  - 


Jx  dx  III 

— .  Ans.  -  log  (x2  +  1 )  -  -  log  (x  4  1)  4  -  tan-1;*;. 

oP  +  xz  +  x  +  1  4  2  2 

f  <fe  2  (a:  +  2)2       3  I 

12.     Apply  the  method  of  Art.  47  to  the  integration  of  - — - — £-. 

(1  +  z)-n-*dz 
The  transformed  expression  is  -  - — g-^- . 

f    x2dx  .        1  xCi  +x2)       1  .      1  +  x 


Examples 

14.  Prove  that 

I  — ; rz  transforms  into 

J  Wi  -x- 


dx                            .            f  (1  +  z)m+»-zdz 
. r-  transforms  into  -     K——± _. 


if  we  make  x  = 


1  +  z 
dx 


dx  .  1     .       .    x         1  x 

l5'     Jsin*(«  +  *cos*)-     ^^a10*81*,"^1^08; 

Multiply  by  sin  x,  substitute  u  for  cos  x,  and  the  integral  becomes 


.6. 


J  3  si 


dx 


n 


3  sin  x  +  sin  ix 


f  ~du 

J  (I -«*)(«  +  **)• 

^«s.  -  log  sin  -  -  log  cos  |  +  -  log  (3+2  cos  x). 

^J  *«-.  (tut 


f     (I-*-)**  V3.     .!  /f+f_2\       »,       /*4  +  *2+.\ 


Let  a;2  =  -,  &c. 

z 

18.  Prove  that 

dx  1 


J  1  + 


—  2  cos  v - 

2n  2n 


log  (  1  -  2#  cos  - —  +  xA 


(ik-  i), 

.   X  —  cos 

.     (2&~  l)ir  I  2» 

+  -  5  sin  i ~-  tan-1  <  ,   , ■— 

2n  .    (2k-  I)tt 


sin 

291  J 

where  &  extends  through  all  integer  values  from  1  to  «,  inclusive. 

f      dx        log  (1  +  a?)         1     „       (2&-1W       /               (2k-i)ir      \ 
19-    5—;  = ; — — Scos- —  log!  i-2zco^ —\x% ) 

7    JI+a;2n+l  2W+I  2W+I  2tl  +  I  6\  2tt  +  I  / 

(2A"-I)7rl 


+ 2  sm  v —  tan"1  < 

2H  +  I  2»  +  1 


2W  +  I 


2»+  I 

where  #  assumes  all  integer  values  from  1  to  n  inclusive. 


(     63 


CHAPTER  III. 

INTEGRATION   BY   SUCCESSIVE   REDUCTION. 

5 1 .  Cases  in  which  sin™  0  cosn  0  d9  is  immediately  In- 
tegrate.— We  shall  commence  this  Chapter*  with  the  dis- 
cussion of  the  integral 

to  which  form  it  will  be  seen  that  a  number  of  other  expres- 
sions are  readily  reducible. 

In  the  first  place  it  is  easily  seen  that  whenever  either  m  or 
n  is  an  odd  positive  integer  the  expression  sinm0  cos"  0^0  can 
be  immediately  integrated. 

For,  if  n  =  ir  +  i,  the  integral  becomes 

Jsinm0  cos2r+10e?0,  or,  j  &mm9  (cos2 9)r d (sin  9), 

If  we  assume  x  =  sin  0,  the  integral  transforms  into 

jxm(i  -xl)rdx\  (i) 

and  as,  by  hypothesis,  r  is  a  positive  integer,  (i  -  x*)r  can 
be  expanded  by  the  Binomial  Theorem  in  a  finite  number  of 
terms,  each  of  which  can  be  integrated  separately.  In  like 
manner,  if  the  index  of  sin  0  be  an  odd  integer,  we  assume 
X  =  cos  0,  &c. 

A  few  examples  are  added  for  the  purpose  of  making  the 
student  familiar  with  this  principle. 


*  It  may  be  observed  tbat  a  large  number  of  the  integrals  discussed  in  this 
Chapter  do  not  require  the  method  of  Successive  Reduction:  however,  sinco 
other  integrals  of  the  same  form  require  this  method,  it  was  not  considered 
advisable  to  separate  the  discussion  into  distinct  Chapters. 


64  Integration  by  Successive  Reduction. 

Examples. 

f                                                                   cos*'  0 
i.       Isin30tf0.  Ans. cos  0. 


2  .          sin50 
-sm30-f . 

3  5 


Icos50rf0.  „     sin0 sin30-f 

Isin^cos7©^. 

J 


COS1O0      cos80 
io    "     8 


/sin5  0^0  i  cos30 

-=E-  »     — „  +  2  cos0  - 


K 


cos20  cos0  3 

2  sin^0      2  sin50 


"  "1         F 


sin30<f0  2  cos^0 


-  2  cos*  0. 


6.  -7=-' 

J  V  cos  0 

fcos30rf0  .  .        3  .  1 

7.  J-^.  „     3Sini0-^sm*0. 

52.  Again,  whenever  m  +  n  is  an  even  negative  integer 
the  expression  sinm  0  cosn  0  d9  can  be  readily  integrated. 
For  if  we  assume  x  =  tan0,  we  have 

;,   sin0  =  j  and  dQ 


v/i  +  x2         yi  +  x*         *  +  «* 

and  the  expression  transforms  into 


a^efo 


(1  +  *2)  2 
Hence,  if  m  +  n  =  -  2r,  this  becomes 

a^ii  +x2)r~1dx, 
a  form  which  is  immediately  integrable. 


Cases  in  which  sinmB  cosn6  dd  is  immediately  Integrable.    65 

Take,  for  example,       tjt-. 

Let  x  =  tan  0,  and  we  get 


\  x2 (i  +  x2) dx,  or  —     -  +  -    — . 


Next,  to  find 


J  si] 


3 

dd 


sin  0  cos50 ' 

Making  the  same  substitution,  we  obtain 

x2)2dx 


P 


x 

Hence,  the  value  of  the  proposed  integral  is 
tan40 


4 

dd 


+  tan20  +  log  (tan  0). 


Again,  to  find    -r-^ ^. 

°  J  sm*0  cos20 


.    (i  +  x2)dx 


Here  the  transformed  expression  is ^ — ,  and  ac 


cordingly  the  value  of  the  proposed  integral  is 


-tan§0 


3  tan20* 

In  many  cases  it  is  more  convenient  to  assume  x  =  cot  0. 

For  example,  to  find   -t-tji. 
r  J  sm40 

dQ 
Since        d(cot0)  =  -  -T-rri,  if  cot  0  =  #,  the  transformed 
v        '        sm20 

integral  is 

-  I  (i  +  x2)dx,  or  -  cot  0 . 

The  following  examples  are  added  for  illustration ; — 
M 


Integration  by  Successive  Reduction. 


f  sitfddd 

J      CO860 

f   de 

J  COS60* 

f         dd 


sin  0  cos3  0 


sini0<?0 
cos*0 


5- 


Examples. 

Jbw. 

tan*0 
4 

to 

2  tan30      tan»0 

tan  0  + +  

3               5 

>» 

tan'0 

-j-  +  log  (tan  0). 

H 

2         3 
"  tan*0. 

» 

8 

-  8  COt  20 COt3  20. 

3 

f  sin*0« 

^      COS*  I 

f         <Z0 

J  sin4  0  cos4  0' 

r        dd  Li-/         tan20\ 

— .  „      2  tan*  0  (  i  +  ) 

J  sini0cos*0  \  5    / 


When  neither  of  the  preceding  methods  is  applicable,  the 
integration  of  the  expression  sinOT0  cos"  0  dd  can  be  obtained 
only  by  aid  of  successive  reduction. 

We  proceed  to  establish  the  formulae  of  reduction  suitable 
to  this  case. 

53.  Formulae  of  Reduction  for  sinm0  cosn  9  dO. 

[  sinm0  cosw0c?0  =  J  cos^fl  sin"1 6d  (sin  0) : 

consequently,  if  we  assume 

H_1Q          sinw+10 
w  =  cosnl0,   v  =    , 

the  formula  for  integration  by  parts  (Art.  21)  gives 

f sin'"0 cosw0^0  -  C°Sn"X 6 Smm+1  °  +  *—  [sin»«deoB"*0rf0.  (2) 
J  m + 1  m+ 1 J 


Case  of  One  Positive  and  One  Negative  Index.  67 

In  like  manner,  if  the  integral  be  written  in  the  form 

-  [sinm-10cosn0^(cos0), 
we  obtain 
f  sin-0  ootf»0rf0=—  [  sin-20  cos«+20  tf0  -  sm""l6lcos?mfl.    (3) 

It  may  be  observed  that  this  latter  formula  can  be  de- 

rived  from  (2)  by  substituting  —  $  f or  0,  and  interchanging 

the  letters  m  and  n  in  it. 

54.  Case  of  one  Positive  and  one  Negative  Index. 

— The  results  in  (2)  and  (3)  hold  whether  m  or  n  be  positive 
or  negative ;  accordingly,  let  one  of  them  be  negative  (n  sup- 
pose), and  on  changing  n  into  -  n,  formula  (3)  becomes 

fsinm0  sin^A         .m-i  fsinm-20   fl        , 

J  cos"0  ™  "  (n  -  1)  cos-1 0  "  n  -  1 J  cos^0  *"'       [    ] 

in  which  m  and  n  are  supposed  to  have  positive*  signs. 

sin^  0 

By  this  formula  the  integral  of »d0  is  made  to  de- 

J  &  cos*  0 

pend  on  another  in  which  the  indices  of  sin  0  and  cos  0  are 
each  diminished  by  two.  The  same  method  is  applicable  to 
the  new  integral,  and  so  on. 

If  m  be  an  odd  integer,  the  expression  is  integrable  im- 
mediately by  Art.  51.  If  m  be  even,  and  n  even  and  greater 
than  m,  the  method  of  Art.  52  is  applicable ;  if  m  =  n,  the 
expression  becomes  J  tanm0<i0,  which  will  be  treated  subse- 
quently ;  if  n  <  m,  the  integral  reduces  to  that  of  sinm_w0  dd. 

— 7^Tn> 
cos     u 


*  The  formulae  of  reduction  employed  in  practice  are  indicated  by  the  capital 
letters  A,  B,  &c. ;  and  in  them  the  indices  m  and  n  are  supposed  to  have  always 
positive  signs.  By  this  means  the  formulae  will  be  more  easily  apprehended 
and  applied  by  the  student. 

[5  a] 


68  Integration  by  Successive  Reduction. 

and  if  n  <  m.  it  reduces  to     7i — .     The  mode  of  find- 

J         COS0 

ing  these  latter  integrals  will  be  considered  subsequently. 

Again,  if  the  index  of  sin  0  be  negative,  we  get,  by 
changing  the  sign  of  m  in  (2), 

fcos"0    a  cos""1 0  n-  1  fcos"-20 

J  sinm0      ~      (m  -  1)  sinm_1 0  ~  m-i  J  sin"*-2©     '        ^    ' 

We  shall  next  consider  the  case  where  the  indices  are 
both  positive. 

55.  Indices  both  Positive. — If  sinm0  (1  -  cos20)  be 
written  instead  of  sinm+2  0  in  formula  (2),  it  becomes 

f    ■   ma        nnjn      cos""1 0  Bin"1"  0 

sm"^  cos"0e?0  = 

J  m  +  1 

cos""1 0  sinm+1 0 


m  +  1 


+  - — !  f  sin"1 0  (cos""2  0  -  cos"  0)  dO  = 
m+ 1 J  v  ' 

+  ?—±  f  sinm0  cos""20tf0  -  — —  [  sinm0  Gosn0dB: 
m+i)  m  +  1  J 

hence,  transposing  the  latter  integral  to  the  other  side,  and 

,.  .,.      ,     m+n  , 

dividing  by ,  we  get 

sin"»0  cos"0 dO  =  -  +  —    sin™0cos"-20rf0.  ((7) 

In  like  manner,  from  (3),  we  get 
f Bm"0  cofl»0rf0  =  —  f  sin-2 0 cos" 0^0  -  S^H^1  . (2>) 


By  aid  of  these  formulae  the  integral  of  sinm  0  cos"  BdO  is 
made  to  depend  on  another  in  which  the  index  of  either 
sin  0,  or  of  cos  0,  is  reduced  by  two.  By  successive  appli- 
cation of  these  formulas  the  complete  integral  can  always  be 
found  when  the  indices  are  integers. 


Indices  both  Negative.  69 

56.  Formulae  of  Reduction  for  sinM  0  dO  and  cosn0dd. 
These  integrals  are  evidently  cases  of  the  general  formulae 
(C)  and  (D) ;  however,  they  are  so  frequently  employed  that 
we  give  the  formulae  of  reduction  separately  in  their  case, 

f         -  ,„     sin0  cos^fl     n-i  [      „  9/1  7n  ,  . 

cosw  BdB  = + cos"-2  0  dB.  (4) 

J  •      n  n    J 

suinddO  = + sinw_20^0.         (5) 

J  w  w     J 

The  former  gives,  when  n  is  even, 

f  <m»0d6  =  —fcos^O  +  -—  cos"-30 
J  w     \  n  -  2 


+    .-,  (—3    cos„-5g  +  &c. 

(W  -  2)(W  -  4) 

+  (^~  Q(^-3)(^-"5)  •  -  ■  ', 
w  («  -  2)  (11  -  4)  ...  2 


(6) 


A  similar  expression  is  readily  obtained  for  the  latter 
integral. 


Examples. 


f  .  ,                      „         sin0  cos 0/  .  „       3\      3 
1.  sin40 <?0.  -4ws. I  sm20  +  -     +  §  0. 

f       „      .  ,     ,  sin0  cos  0/ sin4 0      sin20      i\       0 

2.    ]«*«.,<««.     „  — —  (—  --n--?)+rt- 

f       „                                 sin0cos30/      ,.      <\       </,  \ 

3.  cos60<?0.  ,,     7 (cos20  +  -J +  -^(sm0cos0  +  0j. 

57.  Indices  both  Negative.— It  remains  to  consider 
the  case  where  the  indices  of  sin  6  and  cos  6  are  both 
negative. 

Writing  -  m  and  -  n  instead  of  m  and  n9  in  formula  (C), 
it  becomes 

(■         dO  -  i n+  i  (•  dO  . 

J  sinm0 cosw0  ™  (m  +  w)cosw+,0  sin1"-1©  +  m  +  rcj  sinw0  cosn+20  ' 


70                  Integration  by  Successive  Reduction. 
or,  transposing  and  multiplying  by , 

ft  T    I 

d$ 


f         dd i m  +  n  f 

J  sinm0  cosn+20  ~  (n  +  i )  cosn+10  sinm_10      n  +  i  J  sii 


sinm0cosn0' 

Again,  if  we  substitute  n  for  n  +  2  in  this,  it  becomes 
d0  1 


sinm0  cosn 0      (w  -  1 )  cosn_1 0  sin"1"1 0 
m  +  n  -  2  f         <£0 


+ 


n-i     Jsinm0cos"-20*  v    ' 


Making  alike  transformation*  in  formula  (2)),  it  becomes 

r       dO -I 

J  sinm0cos"0  "  (m-  1)  sin"*-1 0cosn"10 

m  +  n-  2C         dO 


±n-2C__dd__ 

71-1    Jsinm~20cosn0'  v    ; 


+ 
m 

In  each  of  these,  one  of  the  indices  is  reduced  by  two 
degrees,  and  consequently,  by  successive  applications  of  the 
formulae,  the  integrals  are  reducible  ultimately  to  those  of 

one  or  other  of  the  forms  — -  or  -s — ^ :  these   have  been 

cos  v        sin  0 

already  integrated  in  Art.  17. 

The  formulae   of  reduction  for   -. — -n  and  7:    are    so 

smM0  cosn0 

important  that  they  are  added  independently,  as  follows  : — 


*  It  may  be  observed  tbat  formulas  (2?),  (D),  and  (F)  can  be  immediately- 
obtained  from  (A),  (C),  and  {E),  by  interchanging  the  letters  m  and  n,  and 

substituting <p  instead  of  0.     For,  in  this  case,  sin  0,  cos  Q,  and  dd,  transform 


into  cos  <p,  sin  <f>,  and  -  d<p,  respectively. 


Application  of  Method  of  Differentiation.  71 

f   dO    m  sin0  n  -2  f    dO 

J  cosw0  "  (»  -  1)  cos"-^  +  n  -  1 J  cosn-20*  ^7' 

fdO  -  cos0  n  -  2  f    dO  .  . 

sinn  0  =  (w  -  1)  sin*"1 0  +  rc  -  1 J  sin""2  0'  '' 

It  may  be  here  observed  that,  since  sin20  +  cos20  =  1,  we 
have  immediately 

f      dO       =  [__JA__    f       gg 

J  smm6  eosn6  ~  J  sin^G  eosnd  +  J  sinm0  cosw"20  ;  ^ 

and  a  similar  process  is  applicable  to  the  latter  integrals. 
This  method  is  often  useful  in  elementary  cases. 

Examples. 

f        dd  CsmOdO      [   dd  1         ,      t      0 

1.       I  - --  —  \ 1-  I  = 1-  log  tan  -. 

Jsin0cos20      J  cos»0       J  sine      cos  a  2 

f       dd         _  f  sin  Odd      f       dd 
Jsin0cos40      J   cos40       J  sin 0 cos2 0* 

and  is  accordingly  immediately  integrated  by  the  last. 


f   dO  .  cos0        1  . 

3-       J  335.  An*.-— ^  + -log  tan 


2sina0      2     °        a 


J(f0  1  cos0       3  0 

sin3 0 cos2 0*  "    c^siTim2?"1"  2  °g  an? 

58.  Application   of  Method  of  Differentiation. — 

The  formulae  of  reduction  given  in  the  preceding  Articles 
can  also  be  readily  arrived  at  by  direct  differentiation. 
Thus,  for  example,  we  have 

d  /sinm0\      msinm_10     n  sinmfl0 


dO\cosn6J       cosw-J0         cosn+10 

and,  consequently,     x 

fsinm+10       _  1  sin™0      mfsinffl-x0  Q 
J  cosw+10       "  n  cosn0      n  J  cosn_1 0 

This  result  is  easily  identified  with  formula  (A). 


72  Integration  by  Successive  Reduction, 

Again, 
-^  (sni^fl  cosw0)  -  m  sin™-^  cosn+10  -  n  sinm+10  cos"-^. 

If  we  substitute  for  cosn+10  its  equivalent  cosn-10  ( i  -  sin20), 
we  get 

-^  (sin^  cosn0)  =  m  sinm_10  cos""1©  -  (m  +  n)  sinm+10  cosn_10 ; 

hence  we  get 

(.  -***      „  ,/»  7/1         sinm0cosw0        m     f  .  m  ,_       .  ,.  7_ 
sin^cos"-^^  = + sm^flcos"-^^, 
m  +  n          m  +  nj 

a  result  easily  identified  with  (D). 

The  other  formulae  of  reduction  can  be  readily  obtained 
in  like  manner. 

59.  Integration  of  tann0^0  and  - — ^. 

tan  (7 

These  integrals  may  be  regarded  as  cases  of  the  preceding : 
they  can,  however,  be  arrived  at  in  a  simpler  manner,  as 
follows : — 

Since  tan20  =  sec20  -  1,  we  have 

[  tann0  tf0  =  [  tan»"20  (sec20  -  1)  dQ  =  [  tann~20  d  (tan  0) 

-ftan»-20^0  =  ^l0-(tan»-20^0.     (10) 
By  aid  of  this  formula  we  have,  at  once, 

fA    iflJfl    tann-10    tanW_30    tanW_50    st  1    \ 

tann0  dB  = + &c.  (11) 

J  n-  1        n-  3        n  -  5 

( 1.)  If  n  =  2r  +  1,  the  last  term  is  easily  seen  to  be 

(-  i)r+1log(cos0). 
(2.)  If  n  =  2r,  the  two  last  terms  may  be  represented 

by  (-  i)m(tan0-0). 


Trigonometrical  Transformations.  73 

In  a  similar  manner  we  have 

J  tanw0  "  J  tann0       J  tanw"20  ~  (n  -  i )  tanw-!0     J  tan"-20'  l  * 2  j 


Examples. 

f  tanW0. 

,       tan30           „     „ 

Am. tan  6  +  6. 

3 

f    dd 

J  tan3©' 

»  -rt^-log(sin0)- 

J  tan'S* 

- x        x 

"      4tan^  1   2tan20  1  l0< 

f  cot*dd0. 

cot30 

„ +  cot  e  +  e. 

6o.  Trigonometrical  Transformations. — Many  ele- 
mentary integrations  are  immediately  reducible  to  one  or 
other  of  the  preceding  formulae  of  reduction  by  aid  of  the 
transformations   given  in   Art.    26.      For  example,   if  we 

assume  x  =  a  tan  0,  the  expression  -  transforms  into 

(«2  +  #2)5 
sinm0  cosn~m_20  dd  (neglecting  a  constant  multiplier). 

-In  like  manner,  the  substitution  of  a  sin  0  for  x  trans- 

xmdx     .   .    am-n+lsmm0d0         .    .„ 
torms  the  expression   ■ into — ^ :  and,  if 

a  ,,  .        xmdx     ,        £         .  ,     cosn-m-26d6 

x  =  a  sec  a,  the  expression transforms  into : ^ — 

(x*-a>)°  bsh^B 

(neglecting  the  constant  multiplier). 

A  similar  transformation  may  be  applied  in  other  cases. 

Tor  example,  to  find  the  integral  of  - -r, ; 

[2  ax  —  x  j* 

let  x  =  2a  sin20,  then  dx  =  4a  sin  0  cos  6  dd, 

and  the  transformed  integral  is 

2w+1^J'sin2w0tf0: 
accordingly  the  formula  of  reduction  is  the  same  as  that  in  (5) ; 


74  Integration  by  Successive  Reduction. 

Examples. 

Jx*dx                                     I  .  3   .            a;\/ 1  -  rr2 , 
__ .  ^  j-3  m-.,  -  v_ (3  +  „,,. 

C  dx  I  .         I  -  \/l  -X2        */~\  -Z* 

2-   )77f^         "  5log — s sr- 

f       cfo  x  x9 

3'       J  (a2  +  x*)l'  »    a*  (a3  +  a2)*  "  3a«(a2  +"*«)S* 

The  integrals  considered  in  this  Article  admit  also  of 
a  more  direct  treatment.  We  shall  commence  with  the 
following : — 

xm  dx 

6 1 .  Cases  in  which -  is  immediately  inte- 
grahle.                               («  +  erf)* 

We  have  seen,  in  Art.  48,  that  the  proposed  expression  is 
integrahle  immediately  when  m  is  an  odd  positive  integer. 

Again,  when  m  is  an  even  integer,  if  we  assume  a  +  ox2 
=  tf  z2,  the  transformed  expresssion  is 


- 

(«*■ 

-c)    * 

dz 

n 

a 

This  is  immediately  integrahle  when  n  -  m  -  1  is  even 
and  positive,  i.e.  when  m  is  either  an  even  negative  integer, 
or  an  evSn  positive  integer,  less  than  n  -  1. 

For  example,    :    becomes  -  - — -7-* ,     and 

(«  +  C&Y  a~zn-i 

accordingly   is   always   integrahle   by  this  transformation, 
since  n  is  an  odd  integer,  by  hypothesis. 


dx 


(a  +  ex2) 
x2dx 


Binomial  Differentials.  75 

Examples. 

x  (  ex2 

;2\f"  '  a*  {a  +  ex2)*  \        3{a  +  cxZ)', 


f-ig 

J  («  +  «**)*  '     aMa  +  c*2)1   *3      5(«+<a2)) 

r     z3tf# 


z3tf#  -  (2«2  +  3^) 

(a*  +  **)**  "      3  (a2  +  *»)* 


f  dx 

J  xi  (a  +  cx2f 


The  differentials  considered  in  this  Article  are  eases  of  a 
more  general  class  called  binomial  differentials. 

62.  Binomial  Differentials. — Expressions  of  the  form 

xm{a  +  bxn)pdx, 

in  which  m,  n,  p  denote  any  numbers,  positive,  negative,  or 
fractional,  are  called  Binomial  Differentials. 

Such  expressions  can  be  immediately  integrated  in  two 
cases,  which  we  proceed  to  determine  by  transformations 
analogous  to  those  adopted  in  the  preceding  Article: — 

(1).  Let        a  +  bxn  =  z  ;  then  x 


and  dx  =  —  I  —7—  )      dz ; 


i.                    */        x  „x«  7       (z-  a)n      z?dz 
hence         xm{a  +  lxnydx  =  * ~~ ; . 


Consequently,  whenever  is  a  positive  integer,  the 

n 

transformed  expression  is  immediately  integrable   after  ex- 
pansion by  the  Binomial  Theorem, 


76  Integration  by  Successive  Reduction. 

(2).  Again,   if  we  substitute  -  for  x,  the  differential 

y 

becomes 

_  y-«P-m-*(ayn  +  b)*>dy. 

This  is  immediately  integrable,   as    in   the  preceding 
case,  whenever  -  is  a  positive  integer ;  i.  e.  when 

+  p  is  a  negative  integer.     In  this  latter  case  the  inte- 
gration is  effected  by  the  substitution  of  z  for  ax""  +  b. 


Examples. 


x5dx  2(1  +a?)i(a?-2) 

— —  .  Ans. . 

c       dx  x 

<£r  (1  +  jc*)i 


3"        J  s2(i  +  «*)8#  "     "  0       * 

4"     J  *(i +  *»)*'  "   (TT^)i* 

When  neither  of  the  preceding  processes  is  applicable,  the 
expression,  if  p  be  a  fractional  index,  is,  in  general,  incapable 
of  integration  in  a  finite  number  of  terms.  Before  proceed- 
ing with  this  investigation  we  shall  discuss  a  few  simple 
forms  of  integration  by  reduction,  involving  transcendental 
functions. 


63.  Reduction  of 


emxxndx, 


where  n  is  an  integer. 

Integrating  by  parts,  we  have 


f  xnemx      n  f 

xnemx  dx  m  ■ af-'e™ dx.  (13) 

J  m        m)  K  °> 

By  successive  applications  of  this  formula  the  integral 

is  made  to  depend  on      emx  dx,  i.  e. 


on  - 


Reduction  ofjxm  (log  x) ndx.  77 

Cemx 
Again,  to  find    —  dx. 

J  x 

Assuming  u  =  emx,  v  =  -; — — ,   and  integrating  by 

[ft        I  )  x 


parts,  we  have 

[emxdx 


m    Cemxdx  .     . 

^Tj^r-  (I4) 


J     xn         (n  -  i)xV{ 
By  means  of  this  the  integral  is  reduced  to  depend  on 

[emxdx 


The  value  of  this  integral  cannot  be  obtained  in  a  finite 
form;  it  however  may  be  exhibited  in  the  shape  of  an 
infinite  series  ;  for,  expanding  emx  and  integrating  each  term 
separately,  we  have 

[emxdx      ,            mx      mfx2         m?xz         _  .     s 
=  logx  +  —  + 1+ +  &c.          (15) 

J        X  °  I  I  .  2i        I   .    2  .  32  ' 

The  integral  of  a'aPdx  is  immediately  reducible  to  the 
preceding,  since  ax  =  e*l0s°.  Consequently,  by  the  substitu- 
tion of  log  a  f or  m  in  (13)  and  (14),  we  obtain  the  formulae 
of  reduction  for 

faTxndx  and    —  dx. 
Jx" 

In  like  manner  we  have  immediately 

j  e~xxndx  =  -  e-*xn  +  n  j e^x"-1  dx.  ( 1 6) 

64.  Reduction  of      j  xm  (logx)ndx. 

Let  y  =  log  x,  and  the  integral  reduces  to  that  discussed 
in  the  last  Article. 

The  formula  of  reduction  is 

f  xm  (log  xY  dx  =  ^1(1°g^ —  f  xm  (log  x)n~l dx.       (17) 


78  Integration  by  Successive  Reduction. 

Examples. 

J  a  [         a  a?  az      ) 

2.  \z>(logx)-dx.        '  „      ^{(log*)3-!^+l}. 

65.  Reduction  of        fxn  cos  axdx. 

tt  f   «  ,       **  sin  # #      w  f   „  .   .         _ 

Here       xn  cos  axdx  = xn~x  smaxdx; 

J  a  a] 

ngain 

xn"x sm axdx  = + xn  2 cos aa; da\ 

J  a  a     J 

hence 

fa;""1  (##  sin  ax  +  n  cos  ax)      n  (n  -  1)  f  „  a  . 

aJ*  cos  axdx  = * = ' — 1-^  [x11-2  cos  axdx. 
a2  a2     J 

The  formula  of  reduction  for  xn  sin  axdx  can  be  obtained 
in  like  manner. 

Again,  if  we  substitute  y  for  sin*1  x,  the  integral 

/  (sin"1^)"  dx 
transforms  into 

J  yn  cos  ydy, 

and  accordingly  its  yalue  can  be  found  by  the  preceding 
formula. 

Examples. 

1.  a?  cos  xdx.    Am.  x3  sin  x  +  3a;2  cos  x  -  3  .  2 .  x  sin  x  -  3  .  2  .  1  .  cos  x. 

2.  \x*smxdx. 

Am.  -  x*  cos  x  +  4.Z3  sin  x  +  4 .  3  .  x?  cos  a;  -  4  .  3  .  2  .  a;  sin  a;  -  4 .  3  . 2  .  1 .  cos  x. 


Reduction  ofj  cosmx  sin  nxdx.  79 

66.  Reduction  of    jeaxGOSnxdx. 
Integrating  by  parts,  we  get 

fcosw#  eax      n  f 
eax  Q0$nX(jx  m   +  _  \eax  cosn-i^  Sinxdx. 
a  a) 

Again, 

eaxQo$n-lxmixdx 


eax  cos75 


j 

~^x  sm  x      if 

eax  (cosn#  -  (n  -  i)  GO%n-%x%mix)dx 


=  +  - — t-1-   eax  eosn~2xdx  -  -  \eax  cosnxdx 

a  a      J  aj 

substituting,  and  solving  for  j  eax  eosnxdxf  we  get 
eax  cos»'i^  ($  cos  x  +  nsin  x) 


\ 


eaxco&nxdx 

a*  +  n' 


n(n  -  i) 


-V ^  f  eax  cosn~2xdx.  ( 1 8) 

a2  +  IT  J  v     y 


The  form  of  reduction  for  eax  sinnxdx  can  be  obtained  in 
like  manner. 

67.  Reduction  of    J  cos™  #  sin  nxdx. 
Integrating  by  parts,  we  get 


cosw#cosw#     m. 

cosm#  sm  nxdx  «= cos"1"^  cos  nx  smxdx : 

n  n, 


■     COSm' 


replacing  cos  nx  sin  x  by  sin  nx  cos  x  -  sin  (n  -  1)  xf  after  one 
or  two  simple  transformations  we  get 


m     .         7          cosm#cosw# 
cos^smw^aa?  = 


m  +  n 
m 
m 


— —    ao^~lx  sin(w  -  1)  xdx.  (19) 


The  mode  of  reduction  for  cosm#  cos  nxdx,  sinw#  cos  nxdx, 
and  sinm#  sin  nxdx  can  be  easily  found  in  like  manner. 


80  Integration  by  Successive  Reduction. 

Examples. 

e**  sin  x 
4  +  «2  v""  """7~ra(4  +  a8)' 


f   -     •  •   j                       .      ««8ini.     .  „         2e"x 

ea*  &\v*xdx.  Ans. (a  sins  -  2  coss)  +  -7 


2.  I  cos's  sin  4* rfs.  , 
J  t>  12  24 

3.  U"* cofPxdx.  „ (cos's  -  sin 2*  +  2). 

J  5 

68.  Reduction  by  Differentiation. — We  shall  now 
return  to  the  discussion  of  the  integrals  already  considered  in 
Arts.  60  and  61 ;  and  comjnence  with  the  reduction  of  the 

xm  dx 
expression  -. rrr.     This,  as  well  as  other  formulae  of  re- 

r  (a  +  ex2)* 

duction  of  the  same  type,  is  best  investigated  by  the  aid  of  a 
previous  differentiation. 
Thus  we  have 

Y  \xm~l{a  +  cx2f  J  =  (m  -  i)xm~2  (a  +  ex2)*  +      *** 

MX  \  j  \Cl  +  ex 


{a  +  cxz)k 

(m  -  1 )  xm~2  (a  +  ex2)  +  ex™ 
(a  +  cx2)^ 

(m  -  1)  axm~2        mcxm 

+ 


(a  +  cx2)*        (a  +  cx2)*' 
hence,  transposing  and  integrating,  we  obtain 

fafdx     _  ^(a  +  ex2)*  _  (m  -  i)a  f    xm~2dx 
(a  +  ex2)*  mc  mc       J  (a  +  ex2)*'         ^     ' 

By  this  formula  the  integral  is  reduced  to  one  or  more 
dimensions ;  and  by  repetition  of  the  same  process  the  ex- 
pression can  be  always  integrated  when  m  is  a  positive 
integer. 

The  formula  (20)  evidently  holds  whether  m  be  positive 


f     xmdx 
0/}(a  +  cx2)n' 


Reduction  of    -. rr~.  81 


or  negative ;  accordingly,  if  we  change  m  into  -  (m  -  2),  we 
obtain,  after  transposing  and  dividing, 

f        dx         _        (a  +  ex2)*        (m  -  2)  c  f         dx 

J xm(a  +  cx%y* =  ~  (m -  1) ^m-x  "  (m  -  ijaj^"^3 (a  +  ex2)*'    ^2I' 

69.  More  generally,  we  have 

—  {xm~l [a  +  cx2)n]  =  (m-  \)xm~2  (a  +  cx2)n  +  mcxm  (a  +  cx2)n~l 

=  (a  +  ex2)*1'1  { (m  -  1  )axm~2  +  (m  +  2n-  i)cxm] . 
Hence 

\xm  (a  +  cx2)n-'dx  = * f- 

J  (m+  zn-  i)c 

-  ,    (M~1)a        [xm-2  ia  +  ^H dXm        (22\ 

(m  +  211-1)  c]        v  '  x     J 

Consequently,  when  m  is  positive  the  integral  can  he 
reduced  to  one  lower  hy  two  degrees.  If  m  be  negative, 
the  formula  can  be  transformed  as  in  the  preceding  Article, 
and  the  integration  reduced  two  degrees. 

We  next  proceed  to  consider  the  case  where  n  is  negative. 

^  f     xmdx 

70.  Reduction  or     -. — , 

J  {a  +  cx2)n 

m  and  n  being  both  positive. 

_  f     xmdx  f  m  .      xdx 

Here  7 ^-  =   xm~x  -, —. 

){a  +  cx2)n     J         (a  +  cx2)n 

C        XL 

Let  xm~x  =  u,  and       ; 


xdx 


cx~ 
I 


2  (n  -  i)c(a  +  ex2)"-1  ~ 
and  we  get 

f    xmdx -  x™"1 m  -  1     r    «m~2<^       .     , 

J  («  +  c#2)n  "  2~O-i)c(a+C02)n-1  +  2~(w  -  i)cj  (a  +  cx2)n~1'   ^3; 

w 


82  Integration  by  Successive  Reduction. 

By  successive  applications  of  this  form  the  integral  admits 
of  being  reduced  to  another  of  a  simpler  shape.  We  are  not 
able,  however,  to  find  the  complete  integral  by  this  formula, 

T 

unless  when  n  is  either  an  integer,  or  is  of  the  form  -,  where 

r  is  an  integer. 

f         x™dx 
71.  Reduction  of    - - — . 

J  (a  +  2bx  +  cx2)* 

By  differentiation,  we  have 

—  {xm~l{a  +  ibx  +  ex2)*}  =  (m  -  i)xm~2{a  +  ibx  +  ex2)* 

xm~'i  (b  +  cx)     _  (m  -  i )  ax™-2  +  (2m  -  1 )  bxm~l  +  mcxm 
{a  +  2bx+  ex2)*  (a+  2bx  +  ex2)* 

h  [         xmdx  _  xm-l(a+  2bx  +  cx2)i 

J  (a  +  2bx  +  ex2)*  me 

(2m -  1) b  f       xm~ldx  (m-  i)af       xm-2dx 

mc       J  (a  +  2bx  +  ex2)*  mc      J  (a  +  2bx  +  ex2)*'      ^ 

This  furnishes  the  formula  of  reduction  for  this  case :  by 
successive  applications  of  it  the  integral  depends  ultimately 
on  those  of 

xdx  _  dx 

and 


(a  +  2bx  +  ex2)*  (a+  2bx  +  ex2)*' 

These  have  been  determined  already  in  Arts.  9  and  12. 

Again,  the  integral  of  -—-. ; ttt  can  be  reduced  to 

&  xm(a+  2bx  +  cx2)* 

the  preceding  form  by  making  x  =  -. 

z 

72.  The  more  general  integral 

f  xmdx 

J  (a  +  2bx  +  cx2)n 

admits  of  being  treated  in  like  manner. 


Reduction  of = rr— .  83 

J  J  (a  +  2bx  +  cx2)n 

For  if  a  +  ibx  +  ex2  be  represented  by  T,  we  have,  by 
differentiation, 

d  fxm~l\  _  (m  -  i)xm~2      2  (n  -  i)&**  (b  +  ex) 
dx  \  YnZl)  T**  T» 

(m-  i)xm-2  (a+  2bx  +  ex2)  -  2 (n  -  1) xm~^  (b  +  ex) 


(m -  1)  axm~2      2b(m -  n) xm"1      (2n  -  m  -  i)cxm 

"J  ~mn  iVn  rfn 


Hence,  we  get  the  formula  of  reduction 

2{m-n)b    Cxm~1dx 


Cxmdx  _  -  xm~l  2{m-n)b    C, 

J  ~T"  =  {in-m-\)cTn-x  +  {2n-m-i)c) 


T 


(m-  i)a      [xm~2dx 

+ 


(2n  -  m 


%      [xm~2dx       .     . 
TTc\-T^-     (25) 


xm  dx 
By  aid  of  this,  the  integral  of     -^  ,  when  m  is  a  positive 

integer,  is  made  to  depend  on  those  of  -=^  and  -?=-n.     Again, 

x  dx 
it  is  easily  seen  that  the  integral  of  -=^  is  reduced  to  that  of 

dx 

Cxdx      1  r  (b  +  cx)dx      b  C  dx 
J  ~T^  =  c  J        Yn  cj  ~Tn 

-  1 b  [dx 

"  2(n-  i)cTn~l      c)  Tn'  {2  ' 

[6  a] 


84  Integration  by  Successive  Reduction, 

f  dx 

73.  Reduction  of     ; — 


20X  +  cx2)n 


In  order  to  reduoe    -^  we  have 
d  fb+cx\  _    c       211  (b  +  ex)* 

c 


Tn 


Hence 


Jdx 


J'n+i 

2n(ac-  bz) 

2nc     2n(ac-b*)      (2n-i)c 

b  +  cx 

(2n  -  %)e  (  dx 

on  ( nn  —  /)2^  ' 

Tn   '    <>„{„*       7)2\  1  Tn'             V-// 

By  aid  of  this  formula  of  reduction  the  integral  of  -=^  can 

be  found  whenever  n  is  an  integer,  or  when  it  is  of  the  form 

r 

-  (r  being  an  integer). 

f          dx 
74.  Reduction  of      7- — = —, 

'*  J(a  +  boo8x)n' 

when  n  is  a  positive  integer. 

Let  TJ=  a  +  b  cos  a?,  then  —  =  -6sin«,     cos  #  -  — r — . 
dx  b 

Again,  by  differentiation,  we  have 

d  (sin  a?)  _  cos  a;      (n  -  i)b  sin2  a? 

~dx  \JF*\  "  Hn~Zx  +  Wn 

_cosa;      (n-  i)b      (n  -  i)b  cos9a? 

substitute  — = —  for  cos  x  in  the  numerators  of  these  fractions, 
0 


(n-  i)b      n  -  1       2(n  -  i)a 


and  we  get 
d  (sin  a?)  _       1 
dx  \  U^1)  =  bU**  "  Til"-1    '         Jjn  0Tjm-2    ■        jjp 

_  (n  -  i>2  =  -  (n  -  2)      (2* -3)0      (tt-i)(a2-62) 


Reduction  of    -. : rr.  85 


dx 

b  cos  x)n* 


Hence,  transposing  and  integrating,  we  get 


dx 


f  dx  -b&mx  (211  -  3)  a      f  d 

J  U*  =  (»-  i)(a2-b2)U^x  +  (n-  i)(a2-b2)  J  IP 

n  -  2         C  dx 

~  (n-i){a2-b2)]Tr^'  (     ' 

By  this  formula  the  proposed  integral  can  be  reduced  to 
depend  on 


J.- 


dx 


+  b  cos  x' 


the  value  of  which  has  been  found  in  Art.  1 8. 

75.  The  integral  considered  in  the  last  Article  can  also 
be  found  by  aid  of  a  transformation,  whenever  a  is  greater 
than  b,  as  follows : — 

dr  dx 


(a  +  6cos.r)n      i,       ,N       ,  x      ,       ,v   .  4#j 
v  y        j  (0  +  b)  cos-  -  +  (a  -  6)  sm2- 

f  1  +  tan2  -  ]  dx 
(A  cos2-  +  i?sin2-  J       (^4 +  2?  tan2 -J 

(where  A  =  a  +  b,  B  =  a  -  b). 

x       jA 
Next,  assume  tan  -  =    l-=r  tan  #,  then 

2         \-£> 

(  1  +  tan2  - )  dx  =  2    /—  (1  +  tan2<£)  <sfy> : 


Integration  by  Successive  Reduction. 

and  we  get 

x\n  (      A  Y1"1 

i  +  tan2- J  dx  FT  ( *  +  ~» tanV )     d$ 


B        -4nseo2"-> 

2  (B  cos20  +  A  sin2<ft)n-x  d<j> 
{AB)n-h 

Hence,  replacing  A  and  B  by  a  +  b  and  a  -  b,  we  get 
f         fifo  f(a  -  b  cos  2<f>)$-1  d<p 

)(a+bcosx)n~2)        {a2-b*)»-h       '  ^29J 

When  n  is  a  positive  integer,  the  integral  at  the  right- 
hand  side  can  be  found  by  expanding  (a  -  b  cos  20)n-1,  and 
integrating  each  term  separately  by  formula  (4). 

Again,  if  in  (28)  we  make  b  =  a  cos  a,  and  2$  =  y,  we 
obtain 

7 " ^  =    •  2n  1       v1  -  cos  a  cos  y)n~ldy,     (30) 

J  (1  +cosacos#)n     sm2n  xaJ  v  '  w  ' 

where  tan  -  =  tan  -  tan  -. 
2  22 

Hence,  if  we  take  o  and  -  as  limits  for  x,  we  have 
2 

7T  a 

. =   .  .   .       (1  -  cos  a  cosy)n-ldy. 

J0(i +  cosocosic)n     sin2n-1aj0v 

/(«)  dx 


76.  Integration  of 


0  W/fl  +  2bx  +  cxt 


"We  shall  conclude  this  Chapter  with  the  discussion  of  the 
above  form,  where /(a?)  and  <p(x)  are  supposed  rational  alge- 
braic functions  of  x. 

Hf(x)  be  of  higher  dimensions  than  <p{x),  the  fraction 
may  be  written  in  the  form 


/  ( Ct  i  u/Yt 

Integration  of —-  87 

<j>(v)va  +  20X  +  c^2 

Again,  since  Q  is  of  the  fornix  +  qx  +  rx2  +  &c,  the  inte- 
gration of  can  be  found  by  the  method  of 

*/ a  +  zbx  +  ex* 
Art.  71. 

r> 

The  fraction  — r-r  can  be  decomposed  by  the  method  of 

partial  fractions  (Chap.  II.).     To  any  root  a,  which  is  not  a 

A 

multiple  root,  corresponds  a  term  of  the  form ,  and  the 

x  —  a 

corresponding  term  in  the  expression  under  discussion  is 

Adx 

(x  -  a)  */a  +  zbx  +  ex2 

The  method  of  integration  of  this  has  been  given  in  Art.  13. 
Next,  to  a  multiple  root  correspond  terms  of  the  form 

Bdx 


(x  -  a)r\/a  +  zbx  +  ex2 
This  is  reducible  to  the  form  of  Art.   71   on  making 
x  -  a  -  -.     Again,  to  a  pair  of  imaginary  roots  corresponds 
an  expression  of  the  form 

(Ix  +  m)dx 


{ (x  -  a)2  +  /32)  <\A  +  zbx  +  ex2' 
'or  x  -  a,  i 

(Lz  +  M)  dz 
(z2  +  /32)  VA  +  zBz  +~Cz* 


If  z  be  substituted  for  x  -  a,  the  transformed  expression 
may  be  written 


where  L,  M,  A,  B,  C,  are  constants. 

To  integrate  this  form ;  assume*  %  =  )3  tan  (6  +  7),  where 

*  For  this  simple  method  of  determining  the  integral  in  question  I  am 
indebted  to  Mr.  Cathcart. 


88  Integration  by  Successive  Reduction. 

0  is  a  new  variable,  and  7  an  arbitrary  constant,  and  the 
transformed  expression  is 

(Z/3  sin  (0  +  y)  +  Mcos  (0  +  y)\dB 

(3^ Acof?(6  +  y)  +  2Bp  eos(0  +  y)  Bm(8  +  y)  +  Cp  sm\8  +  y)' 

Again,  the  expression  under  the  square  root  is  easily 
transformed  into 

±{A  +  <?j32  +  (A-  Cp2)  cos  2 (0  +  y)  +  2B(5  sin  2(0  +  7)} 
=  %\a  +  C/32  +  cos  2d[(A-  0/32)  cos  2y  +  iB$  sin  2y J 

+  sin  20  {2i?j3  cos  2y  -  (A  -  Cfi2)  sin  27}    . 

Moreover,  since  y  is  perfectly  arbitrary,  it  may  be  assumed 
so  as  to  satisfy  the  equation 

2B[5  cos  27  -  (A  -  <7j32)  sin  27  =  o,  or  tan  27  =     .  _  „ „t : 

and  consequently  the  proposed  expression  is  reducible  to  the 
form 

(Z'cos0  +  M'  amO)dd 

-/P+   QCOS20 

(in  which  L',  M%  P  and  Q  are  constants),  or 
XW(sin0)  If'rf(eos0) 


yP+  Q-  2Q sin20     yP-Q+  2Qcos20* 
each  of  which  is  immediately  integrable. 


Examples,  89 


Examples. 


I.       I  cos30  sin20<£0.     Ans.  - 


2.  * 8  cos3 Ode.         , 


5 
sin30      sin50 


I  sir 

3.  sin50  cos50  <?0.        „    -  —  j  cos  20  -  -  cos320  +  -  cos5  20  j . 

fcos*0<?0  cos30  ,       /        0\ 

4'       J"^0-  "    —  + cose  +  log  ^tonjj. 

fcos40tf0  /      ■        3  \     1         3,  /0\ 

J(i+x2)5  \5-3  3  /(l+S2)* 

J,   ..        ,    ,    _  (a  +  fow)  J>+i  { (p  +  1)  &r»  -  a} 

w(jt?  +  l)(^  +  2)52 

8.       I  r*cos3ar<fr.  „     —  J  3  (sin  a  -  cos  #)  +  cos2#(3sina:-cos3:)(. 

f  dO  A  f  dd 

J  sin™  0  cos"  0  ~  sin»»-10cosn-10  J  sinm"2  0  cos"  0* 


determine  the  values  of  A  and  B  by  differentiation, 
(z2  -  a2)*?* 


f  (a;2  -  a^da 
''     J    (s2  +  a2)3 

f    sin20  <?0 

J  (I+COS0)2' 

Jsinw0  dd  f  sinmd>  «?d> 
-—  transforms  into  2m_nfl  1  — = , 
(I  +  COS0)n                                             J  cos2"-™^ 


.     sin20  dd 
"•      \- — : — •  Ans.  2tan- 


where  0  =  2$. 


90  Examples. 

r        dx 

I3'     J  {a  +  b  cos  *)2" 

.  -  bsinx  2a  ,  (  /a-  £\»        a;) 

^W5-  /I 5T7 1 f  + ;  tan-1     ( )    tan  -   . 

{a*  -  b*){a  +  b  cos x)      (4,  _  4,ji  t  \«  +  V  2) 

f      cosflrffl  (j  sin  0  8  ,  f     n  2  \ 

14.  1  - jjj.        Ans.  -  . tan1!  —   . 

J  (5  +  4  cos  0)2  9      5  f  4  cos  0      27  \    3    / 

15.  I  (sin1*:)4  dx  =  x  { (sin*1  a;)4  -4.3.  (sin_1a;)2  +4.3.2.1} 

+  4\/i  -*2  sin"1  a;  {(sin-1  a;)2  -  3  .  2}. 

z-     t.  1        »  A  .t   .  «    ,  f(co3x)dx 

16.  Prove  by  Art.  74,  that  any  expression  of  the  form   .       , — ' — —  u 

{a  +  b  cos  x)n 
capable  of  being  integrated  when /(cos  x)  consists  of  integral  powers  of  cos  x. 

17.  Show,  in  like  manner,  that  the  expression 

/(cos x,  ainx)dx 
{a  +  b  cos  x)n 

can  be  integrated  when  /(cos  x  sin  x)  consists  only  of  integral  powers  of  cos  x 
and  sin  x. 

C         dx 

+     )a  +  bx  +  cz* 
find  the  values  of  P,  Q,  and  H. 

fdd  (a  +  b)d>  {a  -  b)  sin  2d> 
.  Ans.  > Y-  -  , 
(a  cos?  0  +  3  sin2  0)2                                  2  (a£)*            4  (a£)* 

where  tan  £  =    r  tan  0. 


20.  Find  the  values  of  n  for  which  f     f —  is  integrable  in  finite 

terms. 

2 1 .  Prove  that 


x*dx 
\/a*n  -  x2n 


—  =   .  „   ,  (1  -  cos  a  cos  y)n-x< 

Jo  (1  +  cos  a  cos  x)n      sin2"-1  a  J  0 


(    »1     ) 


CHAPTER  IV. 

INTEGRATION    BY   RATIONALIZATION. 

77.  Integration  of  Monomials. — If  an  algebraic  expres- 
sion contain  fractional  powers  of  the  variable  x  it  can 
evidently  be  rendered  rational  by  assuming  x  =  zn,  where  » 
is  the  least  common  multiple  of  the  denominators  of  the 
several  fractional  powers.  By  this  means  the  integration  of 
such  expressions  is  reduced  to  that  of  rational  functions. 
For  example,  to  find 

(1  +  xi)dx 

I    +  X* 

Let  x  =  s4,  and  the  transformed  expression  is 

z)dz 


(V(i  +  z 

1  +z- 


Consequently  the  value  of  the  integral  is 

AX% 

—  +  2xh  -  ^xi  +  4  tan"1^)  -  2  log  (1  .+  x*). 
o 

Again,  any  algebraic  expression  containing  integral 
powers  of  x  along  with  irrational  powers  of  an  expression 
of  the  form  a  +  bx  is  immediately  reduced  to  the  preceding, 
by  the  substitution  of  z  f  or  a  +  bx. 

Examples. 
Am.  Z2L [5z3  +  6x2  +  u  +  l6-j# 


Wx-i  5-7 

Jxdx  2   (20  +  bx) 

(a  +  bxf  "     b*  */*  +  b* 


J  x  +  y  x  - 1  Y  3  *         V3         ' 


92  Integration  by  Rationalization. 


78.  Rationalization  of  F(x,  </a  +  zbx  +  cx2)dx.  It 
has  been  observed  (Art.  28)  that  the  integration,  in  a  finite 
form  of  irrational  expressions  containing  powers  of  x  beyond 
the  second,  is  in  general  impossible  without  introducing  new 
transcendental  functions.  "We  shall  accordingly  restrict  our 
investigation  to  the  case  of  an  algebraic  function  containing 
a  single  radical  of  the  form  */ 'a  +  2bx  +  ex2,  where  a,  b,  c  are 
any  constants,  positive  or  negative. 

Integrals  of  this  form  have  been  already  treated  by  the 
method  of  Reduction  (Art.  76).  We  shall  discuss  them  here 
by  the  method  of  rationalization. 

■f  (gyi  /Fw 

The  expression*  ^-7-4  ■  ,  can  be  made   ra- 

0W  */a+2bx+cx2 

tional  in   several  ways,   which  we  propose  to   consider  in 
order : — 

(1).  Assume         «/a  +  ibx  +  ex2  =  z  -  x  </c.  (1) 

Then  a  +  zbx  =  z2  -  2xz  ^/c ;  .*.  bdx  =  zdz  -  */c  (xdz  +  zdx)9 

or  dx (b  +  z  a/c)  =  dz(z  -  x  y^c)  =  dz  */a  +  zbx  +  ex2 ; 

dx  dz  .  „ 

(2) 


-y/a  +  zbx  +  c&     b  +  z^/c 

Also  x  =  ~*  (3) 

2(b  +  z*/c) 

This  substitution  obviously  renders  the  proposed  ex- 
pression rational ;  and  its  integration  is  reducible  to  that  of 
the  class  considered  in  Chapter  II. 


*  It  will  be  shown  subsequently  that  the  integration  of  all  expressions  of 
the  form 

F(x,  *y  a  +  zbx  +  ex2)  dx 

is  reducible  to  that  of  the  above  when  F  is  a  rational  algebraic  function. 

It  may  also  be  observed  that,  in  general,  the  most  expeditious  method  of  in- 
tegration in  practice  is  that  of  successive  Reduction  (Arts.  71,  72,  76). 


Rationalization  of  F(x,  ^/a  +  zbx  +  c#2)  da?.  93 

When  b  =  o,  we  get 

c?#             cfe          n          s2  -  a  .        .    .      . 
—  =  — — ,  and  a;  =  —  (see  Art.  9). 

V  «  +  C#2        S^C  22^/0 

By  aid  of  the  preceding  substitution  the  expression 
dx 


(x  - p)  */a  +  zbx  +  ex 
transforms  into 


(Art.  13) 


z2  -  2zp  ^/c  -  a  -  2pb 

f  dx 

For  example,  to  find — . 

J  (p  +  qx)  y  l  +  x* 


z2-  1 


dx  2< 


Here    x  = ,  and 

2Z  (p  +  qX)  */\  +  x*      qf  +  2pz-  q 

.  r dx         =     '   -  io  UzJrP-  yp2+f\ 

J  (p  +  qx)  \/i  +  x2      */p2  +  £2         \qz  +  p  +  */p2  +  q2J 

"When  the  coefficient  c  is  negative  the  preceding  method 
introduces  imaginaries :  we  proceed  to  other  transformations 
in  which  they  are  avoided. 

(2).  Assume*      */ a  +  2bx  4-  ex2  =  */a  +  xz.  (4) 

Squaring  both  sides,  we  get  immediately 

zb  +  ex  =  2z  ^/a  +  xz2 ; 

.\  dx(o  -  sa)  =  2dz(^/a+  xz)  =  2dz  \/a  +  2bx  +  ex2. 

t-t                              dx                 2dz  .  , 

Hence  — ======  = .  (5) 

y/0+2fo?  +  C#2       C-Z2 


*  This  is  reducible  to  the  preceding,  by  changing  x  into  -,  and  then  em- 
ploying the  former  transformation. 


94  Integration  by  Rationalization, 

a    j  2(%\/a-b) 

And  x  =     \V_gZ     >.  (6) 

This   substitution   also   evidently  renders  the  proposed 
expression  rational,  provided  a  be  positive. 
For  example,  to  find 

[•        dx 

J  X^/l   -  X2 

Assume  *y  i  -  x2  =  i  -  xz,  and  we  get 

\7y7^ '  j?  - lo»2  - los  ('"  /"^ 

(3).  Again,  when  the  roots  of  a  +  ibx  +  c#2  are  real,  there 
is  another  method  of  transformation. 

For,  let  a  and  j3  be  the  roots,  and  the  radical  becomes 
of  the  form 

-v/c  (x  -  a) (x  -  j3),  or  </c(x  -  a)(fi  -  x), 

according  as  the  coefficient  of  x2  is  positive  or  negative. 

In  the  former  case,  assume  */x  -  a  =  z  */x  -  /3,  and  we 

get 

a  -  Bz2     .  a  _  3  dx         2zdz 

x  = '— ;  hence  x  -  /3  =  -„ ;   .*.  3  = ,. 

1  -  z2  ^       1  -  z2  x  -  [5      1  -  z2 

Accordingly 

dx  dx  2      dz 


\/c(x-a)(x-l3)      z{x-^)^rc      t/ci-z* 
In  the  latter  case,  let  ^/x  -  a  =  z  ^/j5  -  x,  and  we  get 


(7) 


x  = 


a  +  j3s2 


1+2 


1  9 


,                                  dx                   2      dz 
and  „  =         §  /8) 

\/c(x-a)(fi-x)     */c  i+z2 


Rationalization  of  F(x,  */a  +  ibx  +  ex2)  dx.  95 

For  example,  the  integral 

f  dx 


J  (p+qx)  a/  i  -x2 
transforms  into 


}{p  +  q)z2+p-qi 
z2-  I 


on  making  x 


The  student  can  compare  this  method  of  integrating  the 
preceding  example  with  that  of  Art.  13,  and  he  will  find  no 
difficulty  in  identifying  the  results. 

It  may  be  observed  that  in  the  application  of  the  fore- 
going methods  it  is  advisable  that  the  student  should  in  each 
case  select  whichever  method  avoids  the  introduction  of 
imaginaries. 

Thus,  as  already  observed,  the  first  should  be  em- 
ployed only  when  c  is  positive :  in  like  manner,  the  second 
requires  a  to  be  positive;  and  the  third,  that  the  roots 
be  real. 

It  is  easily  seen  that  when  a  and  c  are  both  negative,  the 
roots  must  be  real ;  for  the  expression 


/ -7 -         lb2  -ac-  (ex  -  b)2 

</-  a  +  2bx  -  ex*,  or    / — 

is  imaginary  for  all  real  values  of  x  unless  b2  -  ac  is  positive ; 
i.e.  unless  the  roots  are  real. 

Accordingly,  the  third  method  is  always  applicable  when 
the  other  two  fail. 

From  the  preceding  investigation  it  follows  that  the 
expression 

F(x,  */a  +  ibx  +  ex2)  dx 

can  be  always  rationalized ;  F  denoting  a  rational  algebraic 
function  of  x  and  of  ^/a  +  zbx  +  ex2. 


Integration  by  Rationalization, 


Examples. 


t  dx  i      ,      \/4  +  2a;-V  2-as 

'. .  Ana.  —^  log  — =. 

J  (2  +  3*)v  4~*2  4V»         -s/4+2^  +  ^2-* 

2*       ]  [(a-  +  x2)i  +  *]*' 
Assume  «  =  (a2  +  a2)*  +  x,  and  we  get  for  the  value  of  the  proposed  integral 


2  A    2  — 

3  ~5^* 


p*  +  *  v^2  +  a;2  -  2 


3.  cte  V  x  +  v  2  +  x2.  Ant. 

J  3     Vaj  +  y/^ 

4.  f  xm  { (a2  +  x2)i  +  x}n  dx. 

Making  the  same  assumption  as  in  Ex.  2,  the  transformed  expression  is 
(z2  -  a2)"  (a*  +z2)dz 

which  is  immediately  integrable  when  m  is  a  positive  integer. 

r  <to  [(i +«»)*  + a?]*1      [(1  -f  s2)*  +  a;]"-1 

5'        J  {(i+**)l-*}»  m'  2(»+l)  +  2(»-l) 

<&? 

>v/a  +  2&r  +  cs2  (\/a  +  2&p  +  ^2  *  %v  ej 

Let  -v/*  +  2^  +  ca;*  ±  x  V  c  =  z»  then,  as  in  Art.  78,  we  get 
dx  dz 

\/a  +  2bx  +  cx2      b±z^/e 

hence  the  proposed  expression  transforms  into 

dz 

&c. 


(b  ±  z\/c) 


General  Investigation.  97 

79.  General  Investigation. — The  following  more 
general  investigation  may  be  worthy  of  the  notice  of  the 
student. 

Let  R  denote  the  quadratic  expression  a  +  zbx  +  ex2 ; 
then,  since  the  even  powers  of  \/R  are  rational,  and  the  odd 
contain  */ . R  as  a  factor,  any  rational  algebraic  function  of  x 
and  of  v  R  can  evidently  be  reduced  to  the  form 

p  +  Q<y~R 
F+Q'yi? 

where  P,  Q,  P/,  Q*  are  rational  algebraic  functions  of  x. 

On  multiplying  the  numerator  and  denominator  of  this 
fraction  by  the  complementary  surd  Pr  -  Qf  vR,  the  deno- 
minator becomes  rational,  and  the  resulting  expression  may 
be  written  in  the  form 

M+N-/R, 

where  M  and  JV  are  rational  functions. 

The  integration  of  Mdx  is  effected  by  the  methods  of 
Chapter  II. 

at  f       ,-  [NRdx 

Also  \N</  Rdx=\—/^', 

which  is  of  the  form 

f  f(x)  dx 


J  <p  (x)  ^/a  +  zbx  +  ex2 

Let,  as  before,  \/a  +  zbx  +  ex2  =  vc(x  -  a)(x  -  |3),  and 
substitute  vt —, ^-=  instead  of  x,  when  the  radical  becomes 

\/c{\-aK'  +  2(fjL-an')z  +  (v-  av)  z2 }  {  \  -  ftA.'  H-  2  (/x~j8aQ  z  +  {v  -  fly')  z2  } 
A'  +  2u'z  +  i/'z2. 

(9) 

Again,  if  the  quadratic  factors  under  this  radical  be  made 
each  a  perfect  square,  the  expression  obviously  becomes 
rational. 


98  Integration  by  Rationalization. 

The  simplest  method  of  fulfilling  these  conditions  is  by 
reducing  one  factor  to  a  constant,  and  the  other  to  the  term 
containing  s2. 

Accordingly,  let 

A  -  aX  =  o,      fi  -  afi  =  o,      fi  -  j3//  =  o,      v  -  fiv  =  o  ; 
Or  fi  =  O,        //  =  O,        A  =  aX,        v  =  fiv. 

On  making  these  substitutions  the  expression  (q)  becomes 
(g-q)«^XV  while  „  =  «V  +  PvV 

X'+vV  X'+vV 

In  order  that  a/-  cAV  should  be  real,  A' and  v'  must  have 
opposite  signs  when  c  is  positive,  and  the  same  sign  when  c 
is  negative. 

It  is  also  easily  seen  that  without  loss*  of  generality  we 
may  assume  X  =  i ,  and  v  =  ±  i . 

/3    2 

Hence,  when  e  is  positive,  we  get  x  = ^-y,  and  when 

a  +  /3s2 
c  is  negative,  x  = — . 

&  i  +  z2 

These  agree  with  the  third  transformation  in  the  preced- 
ing Article. 

More  generally,  when  the  factors  in  (9)  are  each  squares, 
we  must  have 

(ji  -  afi')2  -  (A  -  aX)(v  -  av')  =  o, 

or         ju2  -  \v  +  (A  v  +  vX  -  2/ufi)  a  +  (fi2  -  AV)  a2  =  o,  ( i  o) 

and  a  similar  equation  with  j3  instead  of  a. 

Moreover,  by  hypothesis,  a  satisfies  the  equation 

a  +  2ba  +  ca2  =  O. 


*  For  the  substitution  of  y2  for  — —  transforms 
A. 

aA'  +  fr/V  .         a+ft/2  . 

— = r^-  into  r  ;     .'.  &c. 

A'  +  v'z1  1  +y2 


General  Investigation.  99 

Accordingly  (10)  is  satisfied  if  we  assume  the  constants 
A,  ju,  &c,  so  as  to  satisfy  the  equations 

/uL2-\v  =  a,     Xv  ■\-\v'-2pfi=2b9     f/2-Xv  =  c.       (n) 

Again,  solving  for  f  from  the  equation 

x(X  +  2flZ  +  vV)  =  X  +  2[AZ  +  vz2,  (12) 

we  obtain 


(v  -  Xv)  Z  +  fl-X[/  =  vV  -  Xv  +  (X  v'  +  Xv  -  2flfl)  X  +  (f/2  -  XV) 


=*/a+  2bx  +  ex2.  (13) 

Also,  by  differentiation,  we  gejb  from  (12), 
(X'  +  2/llz  +  vV)  dx=2[p  +  vz-x(fi+  vz) }  dz 

=  2  v  a  +  2bx  +  cx2dz ; 
dx  2dz 


Va  +  2bx+  ex2      X  +  2/ui'z  +  vz2' 


(H) 


Now,  since  we  have  but  three  equations  (11)  connecting 
X,  ju,  &c,  they  can  be  satisfied  in  an  indefinite  number  of 
ways. 

"We  proceed  to  consider  the  simplest  cases  for  real  trans- 
formations. 

(1).  Let  a  be  positive,  and  we  may  assume  v  =  o,  and 
fi  =  o ;  this  gives 

fi  =  v  «,     Xv'  =  2  b,     XV  =  -  c. 

Again,  without  loss  of  generality,  we  may  assume  v'=-  1, 
which  gives 

\  1      \'  -u  2(z^/a-b) 

X  =  -  20,     X  =  e ;  whence  x  =  — ^— ^ = — -> 

c  -  z* 


,  dx  2t 

and 


2* 


^/a  +  2bx  +  ex2      0  -  z 

These  agree  with  the  results  in  (5)  and  (6). 
[Taj 


100  Integration  by  Rationalization. 

(2).  In  like  manner,  if  c  be  positive  we  may  assume 

v  ■  o,  fx  =  o,    and    v  =  1, 
which  gives 

fx  =  \/c,  X  =  -  a,     and    X  =  2b  ; 
z2  -  a  ,  cfa>  ofe 


x  = 


and 


2  (ft  +  2V^)  v  #  +  26a;  +  c#2      J  +  2^/c 


as  in  (2)  and  (3). 

It  may  be  observed  that  since  these  results  do  not  contain 
the  roots  a  and  j3,  they  hold  whether  these  roots  be  real  or 
imaginary;  as  already  shown  in  Art.  78. 

It  is  easily  seen  that  if  we  make  fi  =  o,  and  //  =  o,  we 
get  the  third  transformation. 

80.  If  the  expression  to  be  integrated  be  of  the  form 

f{x)dx 


v/a  +  2bx  +  cx2 


where  f(x)  is  a  rational  algebraic  function  of  x,  it  is  often 
more  convenient  to  proceed  as  follows : — 

The  substitution  of  2  —  for  #  transforms  the  proposed 


ac  -  b2 


into  ^       c'    -.  where  a'= 

ya'  +  cz2  c 

If  the  even  and  odd  powers  be  separated  in  the  expan- 
sion of  /( 2  -  -  J,  it  can  plainly  be  written  in  the  form 

and  the  proposed  integral  becomes 

f   <j>(z2)dz       Cz\P(z2)dz 
J  */d  +  cz2     J  */d  +  cz2 

The  former  of  these  is  rationalized  (Art.  24),  by  making 
yV  +  cz2  =  yz,  and  the  latter  by  making  */d  +  cz2  =  y. 


Case  of  a  Recurring  Biquadratic  under  the  Radical  Sign.  101 

It  may  be  observed  that  in  general  the  expression 
fix")       dx 
0(#2)  a/  a  +  ex* 
is  also  made  rational  by  the  transformation 


^/ a  +  ex2  =  xy. 

81.  Case  of  a  Recurring  Biquadratic  under  the 
Radical  Sign. — As  the  solution  of  a  recurring  equation  of 
the  fourth  degree  is  immediately  reducible  to  that  of  a 
quadratic,  it  is  natural  to  consider  in  what  case  an  Elliptic 
Integral  (Art.  28),  in  which  the  biquadratic  under  the  radi- 
cal sign  is  recurring,  is  reducible  by  the  corresponding  sub- 
stitution. 

Writing  the  expression  in  the  form 

$(x)dx  (j>(x)dx 

*/a  +2bx  +  cx2  +  2 bx*  +  ax*        m  \T\      1  \       ,  /       i\ 

and,  assuming  x  +  -  =  2,  the  radical  becomes  */a%%  +  2bz  +  c-2a; 
x 

,    ,  dx  f       i 

and  also  —  [x  — 

x  \       x 

Consequently,  in  order  that  the  transformed  expression 
should  be  of  the  required  type,  it  is  obvious  that  $  (x)  must 
be  reducible  to  the  form 


In  this  case 
transforms  into 


(-34*-:) 


^/a  +  2bx  +  cx2+  2bxd+ax4 

f(z)dz 
*Saz2  +  2bz  +  c  -  2a 


102  Integration  by  Rationalization. 

In  like  manner,  the  expression 


*/a  +  ibx  +  ex*  -  2  bx*  +  ax4. 


transforms  into  ^  -,     by    the    assumption 


y/az*  - 


i 
x  —  =  z. 

X 


When  b  =  o  the  expression  can  in  some  cases  be  reduoed 
by  assuming  either 


XT  X 


Examples. 


C(x2-l)dx  l+x2  +  y/i  +  z* 

, -•  Ans.  log - 

J  xv  i  +  x4-  x 

C(x2  +  i)dx  xz-  l+  v/i  +  tfi 

J  i  +  z2  */i  +  x*>  y2     \i  +  x*) 

fr+a;8        dx 
J  I  -  x2  \/i  +x* 


i  +  x1        dx  I  v/i  +a^  +  a"v/2 

=5*  "    "7= log ^ 

x4-  */  2  i  -  a?2 


This  and  the  preceding  were  given  by  Euler  {Calc.  Int.,  torn.  4) :  the 
connexion,  however,  of  their  solution  with  the  method  of  recurring  equations 
does  not  appear  to  have  been  pointed  out  by  him. 

f      (**-  »)*»  .       \A*  +  x2  +  I 


a;2  \/^  +  #2  +  1 

z,  &c 
(x2  -  I) 


Let         x2  +  —  =  z,  &c. 
xi 


*/{x2  +  ax  +  1)  (x2  +  fix  +  1)* 

.      \/x2  +  ax  +  1  +  */x*  +0z  + 

Ans.    2  log 

x 

f  <.-*)*  **.*»-(•). 


Examples.  103 

[    xdx  .3  [2b*  -  3») 

8.  t r-7-..  -4«s.  — 77^ — :  (a  +  bxf. 

J  (a+bx)i  10b2  ' 

^  rl_+_x>  dx _I_  kg  y/JT^±^±Xy/ 3 

Ji  _  ^y'l  +«*  +  «*  '     v/3  1 -a;2 

I0'      J  (1  +  *4){(i  +  *«)i  -  a;2}*'  "      Sm"  \(i  +  z^jl)  ' 

Assume  x  —  ( 1  +  ic4)!  sin  0,  &c. 

11.         i .      „      surV 1. 

J  (1  +  s2»){(i  +a;2")"  -  z2}*  \(i  +  *2»)2V 

I2-      J  (1  +  *)i  +  (x  +  X)V 
Assume  1  +  x  =  z6. 

I3'     J  (i  —  «*)(i  +«*)*" 

-4w*.  — —  log  ^ i —  + tan"1  ■ — -. 

4/2  1  -  x2  4a/2  xy  2 

C(i  +  xi)dx  x 

J    (i-«*)»  (1  -«*)» 

I5'       J        I-X*      ' 

A          T      ,     A/l  +  a;4  4  ary^N        1            ,    x^/z 
-4«s.  —   ~  log    Z _  +  — —  tan-  — - . 

iyj         \  1 -a;2  /    2^2  v/l+^4 

1  -  x2  dx 


.6.     f_! 

J    I    + 


•7.     p 

J     1    + 


2aX+X*    ^/1  +  2ax  +  2^2  +  2az3  +  & 

i  -  «#3  <£k 


ax2  \/i  +  2c#2  +  a2  z4 

1  ,      #  a/2  (c  -  a)  4-  a/  1  +  2C#2  4-  »*  a;4 

/~7 :  log - ,  when  c>  a. 

\/ 2(0-0)     ■  1  +  ax2  ' 

(x</2{a-cy 


1  .    .  /x^/2(a-c)\        , 

„     — ,  sin-1  (  — ~ —rt —  ] ,  when  a 

*/%  («  *  0)  V     I  +  «*-     / 


(     104    ) 


CHAPTER  V. 

MISCELLANEOUS    EXAMPLES   OF    INTEGRATION. 

w   M          „          ^    M  cos  #  +  2?  sin  #  +  C)dx 
82.  Integration  of    a ; — : - — . 

a  cos  x  +  b  sin  x  +  c 

Let  «  cos#  +  b  sin  #  +  c  =  u,  then  -  a  sin  x  +  b  cos x  =  — . 
Next  assume 

A  cos  #  +  B  sin  a?  +  O  =  Xu  +  a  —  +  v, 

and,  equating  coefficients,  we  have 

-4  =  \a  +  fib,    B  =  \b  -  fia,     C  =  \c  +  v. 

Solving  for  A,  p,  v,  we  get 

_  Aa  +  Bb         _  Ab-Ba  (Aa  +  Bb)  c 

A~     a2  +  b2'    M"    «2  +  &2   '     V~C~      <f  +  b2     ' 

„  f  (A  cos  a;  +  J5  sin  x  +  C)  dx 

Hence  * —. — 

J        a  cos  x  +  0  sin  x  +  c 

(Aa  +  2?&)  x     Ab  -  Ba  ,      .  .    . 

*     a2  +  &2     +  ~¥T¥~ log  ("  cos*+  *  Bmx  +  c) 


(a2  +  b2)C-(Aa  +  Bb)c 


C-  (Aa  +  Bb)cC dx 

a2  +  b2  J  a  cos  x  +  b  sin  x  +  c 


The  latter  integral  can  be  readily  found ;  for,  if  we  make 
a  =  r  cos  a,b  =  r  sin  a,  we  get 

a  cos x  +  b  sin  x  =  r  (cos #  cos  a  +  sin  #  sin  a)  =  r  cos  (x  -  a). 


-r  ,        ,.        /.    ficosx,  smx)  dx  mAm 

Integration  of  -^ ; — r- * .  lOo 

a  cos  x  +  b  sin  x  +  c 

On  making  x  -  a  =  6,  the  integral  reduces  to  the  form  con- 
"  in  Art.  18. 
As  a  simple  example,  let  us  take 


f  (A  +  B  tan  x)  dx 

A    A.     i 

Here 


J       a  +  b  tan  x 
A  +  JB  tan  x     A  cos  x  +  B  sin  x 


a  +  b  tail  a?        a  cos  x  +  b  sin  x  ' 
and  we  evidently  have 
f  (A  +  B  tan  x)  dx     (Aa  +  Bb)  x     Ah  -  Ba 


a  +  b  tan  x  a2  +  b2  a2  +  b2 

/(cosx,  sinx)dx 


log(«cos#  +  5sin#). 


83.  Integration  of 


a  cos  x  +  b  sin  x  +  c  ' 


where /is  a  rational  algehraic  function,  not  involving  frac- 
tions. 

As  in  the  preceding  Article,  assume  x  =  6  +  a,  and  the 
expression  becomes  of  the  form 

(j>  (cos  9,  sin  0)  d9 
A  cos  6  +  B 

Again,  since  sin2  6  =  1  -  cos2  0,  any  integral  function  of  sin  6 
and  cos  6  can  be  transformed  into  another  of  the  form 

(pi  (cos  6)  +  sin  0  $2  (cos  0). 

Accordingly,  the  proposed  expression  is  reducible  to 

^(cos  6)d0      <p2(cos  6)  sin  6 d9 
AcosB  +  B         A  cos  0  +  B 

The  latter  is  immediately  integrable,  by  assuming 

A  cos  6  +  B  =  z. 

To  integrate  the  former,  we  divide  by  A  cos  0  +  B,  and 
integrate  each  term  separately. 


106  Miscellaneous  Examples  of  Integration. 

84.  Integration  of 

/(cos  x)dx 

(«i  +  bx  cos  x)  (a2  +  b2  cos  x) .  . .  .  {an  +  bn  cos  x)' 

where/,  as  before,  denotes  a  rational  algebraic  function. 
Substitute  z  for  cos  x  and  decompose 

m 


{ax  +  blz)(a2  +  b2z)  ....(«„  +  bnz) 

by  the  method  of  partial  fractions :  then  the  expression  to  be 
integrated  reduces  to  the  sum  of  a  number  of  terms  of  the  form 

dx 


A  +  B  cos  xf 
each  of  which  can  be  immediately  integrated. 

Examples. 


,.      [_* ,    ^.J.log/L±4^)-Ato-1fci). 

J  cos x (5  +  3  cos x)  10       \i—  sin  a;/     10  \  2    / 

2*       i   •  2   / 1 r»  wnen  «  >  *• 

J  sin2  a;  (a  +  £  cos  a;) 

J -a  cos  a;  £2  ,  /J  +  ascosaA 

Ans. cos-1  [ ) . 

(a2-*2)  sin  a;      (a*-*2)*  \a  +  *cosa;/ 

f  dx  .       tana;      £,  /ir     x\     b2  C       dx 

3-        I — ,    ;    ,  , r.     Ans. rlogtan    -+-)+-  I . 

J  cos2 x (a  +  b cos x)  a        a2    °         \4     2/     a*  J  a  +  bcosx 

85.  Integration  of     {/{x)+f(x)}exdx. 

The  expression  e*Pdx  is  immediately  integrable  whenever 
P  can  be  divided  into  the  sum  of  two  functions,  one  of  which 
is  the  derived  of  the  other. 

For,  let  P  =  f(x)+f{x), 

then  j  e*Pdx  =  J"  <Pf(x)  dx  +  j  &f(x)  dx. 


Differentiation  under  the  Sign  of  Integration.  107 

Again,  integrating  by  parts,  we  have 

J  exf(x)  dx  -/(»)  ex  -  J*  e*f  (x)  dx. 
Accordingly, 

J I/W+/W}  **-</(•)• 

For  instance,  to  find 


i'^ 


-riOX. 

xY 


Here 


(i  +  x)z      i  +  x      (i  +  xy 


er 
consequently  the  value  of  the  proposed  integral  is . 

I     T  X 


Examples. 

[.       \  e*  (cos  x  +  sin  x)  dx.  Am.  ex  sin  x. 

f      i  +  x  log  x  „  - 

a;2  +  i  a;  -  r 


3*  #  +  i 


f       *2  +  '  ., 

4-     H^Tp)** 


86.  Differentiation  under  the  Sign  of  Integra- 
tion.— The  integral  of  any  expression  of  the  form  <p{x,  a)dx, 
where  a  is  independent  of  x,  is  obviously  a  function  of  a  as 
well  as  of  x. 

Suppose  the  integral  to  be  denoted  by  F(x,  a),  i.  e.  let 

F(x,  a)  =  /  (j>(x,  a)  dx, 


108  Miscellaneous  Examples  of  Integration. 

Again,  differentiating  both  sides  with  respect  to  a,  we 
have,  since  x  and  a  are  independent, 

<P .  F(x,  a)  _  d  .  <p(x,  a) 
dadx  da        9 

or  (Art.  119,  Diff.  Calc), 

—  (d'F(x>*)\  _  d  .  <p(x,  a) 
dx\        da       J  da 

Consequently,  integrating  with  respect  to  x,  we  get 

d  .  F(x,  a)  _  Cd  .  0(ar,  a) 
da  )         da  ' 

In  other  words,  if 

w  =  J"  0  (a?,  a)  dx. 

..  du      Cdd>  _ 

then  —  =    -p-  ok, 

aa     J  aa 

provided  a  he  independent  of  x ;  in  which  case,  accordingly,  it 
is  permitted  to  differentiate  under  the  sign  of  integration. 

By  continuing  the  same  process  of  reasoning  we  obviously 
get 

d»u      [*>*(x,a) 

da«~)       dan       ^  W 

where  u  =  \4>{x,  a)dx,  a  being  independent  of  x. 

For  example,  if  the  equation 


J 


e9* 
eP*dx  =  — 
a 


Integration  under  the  Sign  of  Integration.  109 

be  differentiated  n  times  with  respect  to  a,  we  get 

J'-*-(0(t) 

(See  Art.  49,  Diff.  Oalo.). 

Again,  in  Art.  21  we  have  seen  that 

(eP*  (a  sin  mx  -m  cos  mx) 
e°*  sin  mx  ax  =  — s : : -, 
m?  +  or 


Accordingly, 


f. 


.  d  \n  (e0*  (a  sm  mx-m  cos  mx) 
e°*  sm  ma?  efe  =  [  — 


We  now  proceed  to  consider  the  inverse  process,  namely, 
the  method  of  integration  under  the  sign  of  integration. 

87.  Integration  under  the  Sign  of  Integration. — 

If  in  the  last  Article  we  suppose  <j)(x,  a)  to  be  the  derived 
with  respect  to  a  of  another  function  v,  i.e.  if 


,  dv 


then  v  =  j<f>{x,  a)  da. 

Also  by  the  preceding  Article  we  have 


Hence  \v dx  =   F(x,  a)  da. 


(x}  a)  dx  =  F(xy  a) 


In  other  words,  if 


F(x,  a)  =  \(j)  (x,  a)  dx, 


110  Miscellan eous  Examples  of  Integration . 

then  F(x,  a) da  =     [J  0  (x,  a)  da]  dx.  (3) 

It  may  be  remarked  that  the  results  established  in  this 
and  in  the  preceding  Article  are  chiefly  of  importance  in 
connexion  with  definite  integrals.  Some  examples  of  such 
application  will  be  given  in  the  next  Chapter. 

88.  Integration  by  Infinite  Series. — It  has  been 
already  observed  that  in  most  cases  we  fail  in  exhibiting  the 
integral  of  any  proposed  expression  in  finite  terms.  In  such 
cases,  however,  we  can  often  represent  the  integral  in  the 
form  of  a  series  containing  an  infinite  number  of  terms. 

An  example  of  an  integral  exhibited  in  such  a  form  has 
been  given  in  Art.  63. 

The  simplest  mode  of  seeking  the  integral  o£f(x)dx'm  the 
form  of  an  infinite  series  consists  in  expanding  f(x)  in  a 
series  of  ascending  powers  of  x,  and  integrating  each  term 
separately :  then  if  the  series  thus  obtained  be  convergent,  it 
represents  the  integral  proposed. 

It  can  be  easily  seen  that  if  the  expansion  oif(x)  be  a 
convergent  series,  that  of  jf(x)  dx  is  also  convergent. 

For  let 

f(x)  =  a0  +  axx  +  a2x2  +  .  .  .  anxn  +  &c, 
then 

J.,  »  ,                 ayx7,     a^x*               anX"*1 
fix)  dx  =  a0x  + +  +  .  .  .  +  — +  .  .  . 
23                  n+  1 

Now  (Diff.  Calc,  Art.  73),   the  expression  for  f(x)  is 

a  x 
convergent  whenever    —  is  less  than  unity  for  all  values 

of  n  beyond  a  certain  number ;  and  the  latter  series  is  con- 

yi      ax 
vergent  provided —  be  less  than  unity,  under  the  same 

conditions. 

Accordingly,  the  latter  series  is  convergent  whenever  the 
former  is  so. 


Integration  by  Infinite  Series.  Ill 


Examples. 

T    .    1         C    -rl6 

+  &C. 


f      dx        _  *      I  ij ;      IjJ  ^      i  .  3  •  5  £ 
Jy^i  —  a;5      i      26      2.411       2.4.6  il 

f      As  /■-. —  /         1  sin2#      1  .  3  sin4#  \ 

2.        1  .  =  2  \/  sin  a;  1  r  + +■ +...). 

J  y  sin  a;  \         25         2.4     9  / 

f                p                        / 1       pc     xn         p(p  -  g)c2     x2n         .     \ 
2.  (I  +  cxn)qxm'ldx  =  xm    —  + +  £^- — ^  — ■ +  &c.    . 

89.  Expansion  of  log  (1  +  2mcoa%  +  m2)  dx. 

We  shall  conclude  by  showing  that  the  integral 

log  (1  +  2m  cos  x  +  m2)dx 

can  be  exhibited  in  the  form  of  an  infinite  series. 
For  we  have 

1  +  2m  cos  x  +  m2  =  (1  +  mex  _1)(i  +  mer**-1). 

Hence 

log  (1  +  2m  cos  ^  +  m2)  =  log  (1  +  m^1)  +  log  (1  +  me~x>J~l) 

.     =  m  {e**-1  +  e~x"Zl)  -  —  (e2**^  +  e2x^)  +  &o. 

(  m2  ms  0    \ 

=  2  [m  cos  a? cos  2X  +  —  cos  30  -  &c.  . 

\  2  3  J 

Accordingly 

fi     /                        2\j       (      •            2sin2#       „ sin 337   \ 
\log(i+2mco8x+m2)dx=2lmsmx-m2 — 5—  +  mz — 5 J.  (4) 

This  series  becomes  divergent  when  m  is  greater  than 
unity.  In  that  case,  however,  the  corresponding  series  can  be 
easily  obtained. 


112  Miscellaneous  Examples  of  Integration . 

For         i  +  2m  cos  x  +  m2  =  m2 1  i  + ] (  i  + ), 

V         »  J\        ™  J 
and  accordingly 

,        ,  ov  ,  /cOSiC      COS  2X     COS  IX      p     \ 

log(i  +  2mcosx  +  m2)  =  2loe;m  +  2[ —  + — ^--&c.  . 

6V  '  6  V  m        2m2       3m3  J 

Consequently,  when  m  >  1 ,  we  have 

(\      ,  2\  7         1  fsmx    sin  2x    sin  3a;        \ 

log(i  +  2mcos#+w2)a#=2#lo2rm+2 r-—+    a    .  -...  . 

J  '  °         \  m       22m2      3W         J 

From  the  ahove  it  is  easily  seen  that  the  integral 

Jlog(i  +  acosx)dx 

can  be  exhibited  in  the  form  of  an  infinite  series  when  a  is 

less  than  unity :  for  making  a  = we  have 

J  &  1  +  m2 

log  (1  +  a  cos#)  =  log  (1  +  2m  cos  x  +  m2)  -  log  (1  +  m2). 
The  relation  between  m  and  a  admits  of  being  exhibited 
in  a  simple  form ;  for  let  a  =  sin  a,  and  we  get  m  =  tan  -. 
Making  this  substitution  in  (4),  we  get 

log  (1  +  sin  a  cos  x)  dx  =  2x  log  ( cos  -  1 

(".      a  .           ,     ,  a  sin  2X      0    \  ,  . 

tan  -  sin  x  -  tan2 —  +  &c.  1.  (5) 


Examples. 


113 


Examples. 


1(2  cos  x  +  3  sin  x)  dx            .       12a;       5.      , 
). 1-= — ?— * — .        -4«s- —  log  (3  cos  x  +  2  sin  #). 
3  cos  x  +  2  sm  a?                        13       13 


in 

r  <?*  (xa  - 
1         (i 

J  sin  20  -sin 0      6 


sin40 
e*  (a;3  +  x  +  i)dx 


„    -tan0  +  - — -^  tan-1  (tan  0^/2). 
V  1 


2 
efx 


S7T 


log  ( 1  +  cos  0)  +  -  log  (1  -  cos  0)  -  -  log  (1  -  2  cos  0) . 


sin-  tan  -t 
2        2 


y  2  sin  -  +  1 


1  +  sin 


■r*   \v: 


6.  "When  x%  <  1,  proye  that 


r      dx      _x     1  **  ,  1  .  3  » 
J  V^Thm?      i      25      2.49 


.  3  *9      1-3-5  f^ 
2  .  4  .  6  13 


+  -  -  .; 


and  when  x2  >  1 


r       <fa    _   =     1      1    1       1  .  3    1       1-3-5     1 
J  -v/i  +  a4        a;      2  5a;5      2      4  9a;9      2.4.6  13a;13 
7.  Prove  that 


a  b  +  x         a2     (b  +  x) 


C    eax     ,  .(.      ,,       .      a  b  +  x        a 

and  determine  when  the  series  is  convergent,  and  when  divergent. 
8.  Prove  that 

I sin'*  adoo  = — —  + 


/*+  1 


I  .2  ^  +  3 


(A2+l2)(A8  +  32)  SinM+5q, 

1.2.3.4       ,1  +  5 


Substitute  »  for  sin"1  a;  in  the  expansion  of  eKsm  lx(Dif.  Calc,  Art.  87),  &c. 


snv  (a  da 


sin^w      \  (A.2  +  22)  sin^c 


1     /*  +  2 


2.3      /*  +  4 


\(A2  +  22)(\2  +  42)sin^Pa> 
1.2.3.4.5      /*  +  6    "* 


[8] 


(     114    ) 


CHAPTER  YI. 


DEFINITE     INTEGRALS. 


90.  Integration  regarded  as  Summation. — We  have  in 
the  commencement  observed  that  the  process  of  integration 
may  be  regarded  as  that  of  finding  the  limit  of  the  sum  of 
the  series  of  values  of  a  differential/ (x) dxt  when  x  varies  by 
indefinitely  small  increments  from  any  one  assigned  value  to 
another. 

It  is  in  this  aspect  that  the  practical  importance  of  inte- 
gration chiefly  consists.  For  example,  in  seeking  the  area  of 
a  curve,  we  conceive  it  divided  into  an  indefinite  number  of 
suitable  elementary  areas,  of  which  we  seek  to  determine  the 
sum  by  a  process  of  integration.  Applications  of  finding 
areas  by  this  method  will  be  given  in  the  next  Chapter. 

We  now  proceed  to  show  more  fully  than  in  Chapter  I. 
the  connexion  between  the  process  of  integration  regarded 
from  this  point  of  view  and  that  from  which  we  have  hitherto 
considered  it. 

Suppose  0  (x)  to  represent  a  function  of  x  which  is  finite 
and  continuous  for  all  values  of  x  between  the  limits  X  and  x0 ; 
suppose  also  that  X  -  xQ  is  divided  into  n  intervals  xx  -  x0, 
xi  -  xlf  x3  -  x2f  .  .  .  X  -  xn_i;  then  by  definition  (Diff.  Calc, 
Art.  6),  we  have 

Xi-X0  ' 

in  the  limit  when  Xi  =  xQ ;  accordingly  we  have 

0(#O  -  #M  =  [xx  -  0j>)(0'(<ro)  +  fo), 


Limits  of  Integration.  115 

where  e0  becomes  infinitely  small  along  with  xl  -  x0.     Hence 
we  may  write 

0  W  -  <i>  W  -  (^i  -  #<>)  {^'(«o)  +  to) , 
0  (*•)  -  ^  (*0  ■  fa  -  »J  to'O&i)  +  £i) » 

</>  (a?8)  -  0  (a?a)  =  (»s  -  <fc)  W(&  +  e2}, 


0  (X)  -  0  («fc_i)  =  (X  -  aw)  (0>„_i)  +  f«-i)> 

where  £0»  d  .  .  •  e»-i  become  evanescent  when  the  intervals  are 
taken  as  infinitely  small. 
By  addition,  we  have 

<p  (X)  -  <j>  (x0)  -  (»i  -  #o)  0'(«o)  +  («2  -  #i)  0'(a?i)  +  •  .  . 

+  (X  -  a?»_i)  0'(ar«-i)  +  (#i  -  x0)  *o  +(a%  r^i)  fi  + . . .  +  (X-  a?„_i)  cn-i. 

Now  if  rj  denote  the  greatest  of  the  quantities  s0,  ci, .  .  .  £M-i, 
the  latter  portion  of  the  right-hand  side  is  evidently  less 
than  (X  -  x0)  r/ ;  and  accordingly  becomes  evanescent  ulti- 
mately (compare  Din\  Calc,  Art.  39). 

Hence 

<p  (X)  -  <j>  (x0)  =  limit  of  [(a?!  -  x0)  <j>'(x0)  +  (#2-  #1)  #'(#i)  +  •  •  . 

+  (X-xn_1)<p'(xn_1)l,    (1) 

when  w  is  increased  indefinitely. 

This  result  can  also  be  written  in  the  form 

(f>  (X)  -  $  (x0)  =  S0'(a>)  dfc, 

where  the  sign  of  summation  2  is  supposed  to  extend  through 
all  values  of  x  between  the  limits  x0  and  X. 

91.  Definite  Integrals,   limits    of  Integration. — 

The  result  just  arrived  at,  as  already  stated  in  Art.  31,  is 
written  in  the  form 

f(X)-f(x0)  =  \Xf(x)dx,  (2) 

where  X  is  called  the  superior,  and  x0  the  inferior  limit  of  the 
integral. 

[8  a] 


116  Definite  Integrals. 

Again,  the  expression 

x 

dx 


is  called  the  definite  integral  of  <j>(x)dx  between  the  limits  x0 
and  X,  and  represents  the  limit  of  the  sum  of  the  infinitely 
small  elements  0  (x)  dx,  taken  between  the  proposed  limits. 
From  equation  (i)  we  see  that  the  limit  of 

(*i  -*o)/W  +  {*%  ~  *0/(*0  +  •  •  •  +  (x  '  *n-i)/0»-i), 

when  ti  -  x0,  x2-  xly  .  .  .  X  -  #«_i  become  evanescent,  is  got 
by  finding  the  integral  of  f(x)  dx  (i.  e.  the  function  of  which 
f(x)  is  the  derived),  and  substituting  the  limits  x0,  X  for  x  in 
it,  and  subtracting  the  value  for  the  lower  limit  from  that  for 
the  upper. 

If  we  write  x  instead  of  X  in  (2)  we  have 

/(*)-/(*)-(" /(«)*.  (3) 


in  which  the  upper  limit*  x  may  be  regarded  as  variable. 
Again,  as  the  lower  limit  x0  may  be  assumed  arbitrarily,  f(x0) 
may  have  any  value,  and  may  be  regarded  as  an  arbitrary 
constant.     This  agrees  with  the  results  hitherto  arrived  at. 

In  contradistinction,  the  name  indefinite  integrals  is  often 
applied  to  integrals  such  as  have  been  considered  in  the  pre- 
vious chapters,  in  which  the  form  of  the  function  is  merely 
taken  into  account,  without  regard  to  any  assigned  limits. 

As  already  observed,  the  definite  integral  of  any  expres- 
sion between  assigned  limits  can  be  at  once  found  whenever 
the  indefinite  integral  is  known. 

A  few  easy  examples  are  added  for  illustration. 


*  The  student  should  ohserve  that  in  (3)  the  letter  x  which  rtands  for  the 
superior  limit  and  the  x  in  the  element  f'{z)  dx  must  he  considered  as  heing 
entirely  distinct.  The  want  of  attention  to  this  distinction  often  causes  much 
confusion  in  the  mind  of  the  heginner. 


£ 


Elementary  Examples.  117 

Examples. 


rr»  dx.  Am. 


n  +  i 


f  4  sin  0  dd 

2-    Jo "^7-  "  ^2-r- 

3*  / /=•  »     ~\/a  (v/2  -  I). 

Jo  vn«  +  V«  3  v      yv  ' 

r    d%  *_ 

4*       Jo  a2  +  s2'  "     2a 

f«       (fa; 
Jo  x/a*-x* 

6.       1   era*  rfa;  (a  positive). 
Jo 

.     fl * . 

JO   I  +  2X  COS  <J>  +   X2 

dx 


w 

N 

2* 

I 

»» 

a 

<P 

>j 

2  sin^> 

<*> 

>5 

sin  </> 

w 

»» 

a2  +  w2' 

a 

"     a2  +  m2' 


8#    r 

Jo     I    +    2X  COS  <£  +  x2' 

9.       I   e~°*  sin  mx  dx. 

10.  I    «""*  cos  mx  dx. 

Jo 

11.  I       ; s  =  — -  ,  when  ac  -  b2  is  positive. 

J.,  «  +  2te  +  «c2     ^/ae-b* 

92.  To  prove  that 

T  a*-*(i  -  x)m~ldx  m  f  a-->(i  -a?)""1  dx  =    '  -2- 3- -•(»>- 1) 

J0  Jo        v        ;  n(n+i)..(n  +  m-iy 

ichen  m  and  n  are  positive,  and  m  is  an  integer.      /^wf 

4J& 


dx. 


118  Definite  Integrals. 

The  first  relation  is  evident  from  (31),  Art.  32. 
Again,  integrating  by  parts,  we  have 

fs»-l(i  -  x)m-*dx  -  £  (1  -x)""  +  2^-±  L»(i  -  x)' 

Moreover,   since  n  and  m  -  1  are  positive,  the  term 
a^(i  -  a?)m_1  vanishes  for  both  limits  ; 

■\  [  ^(i-x^dx  =  ^— ?  [V(i  -#)m~2^. 

The  repeated  application  of  this  formula  reduces  the  in- 
tegral to  depend  on     xm*nr~2dxi  the  value  of  which  is . 

Jo  m  +  n  -  1 

Hence  we  have 


»  («  +  1)  .  ..  .  (n  +  m-  1) 


I 

This  formula,  combined  with  the  equation 
J  x»-1  (1  -x)m~l  dx  =  I  aP->  (1  -  «)***>, 


shows  that  when  either  m  or  ft  is  an  integer  the  definite 
integral 

xn~l{i  -x)m-ldx 
0 


I 


oan  be  easily  evaluated. 

When  m  and  n  are  both  fractional,  the  preceding  is  one 
of  the  most  important  definite  integrals  in  analysis. 

We  purpose  in  a  subsequent  part  of  the  Chapter  to  give 
an  investigation  of  some  of  its  simplest  properties. 


Examples. 


'•       I    a*{i-x)*dx.  Ans.    £ 

;o  .>  -  7  ♦  11 

I.       \  «*(i  -z)idx. 


'■3 


Elementary  Examples.  119 


93.  Values  of 


smn  xdxa,vkA 


GO$nxdx. 


One  of  the  simplest  and  most  useful  applications  of 
definite  integration  is  to  the  case  of  the  circular  integrals 
considered  in  the  commencement  of  Chapter  III. 

We  begin  with  the  simple  case  of 


&mnxdx. 


If  in  the  equation  (Art.  56) 

f  .  m     .  cos  #  sin"'"1  a?     »-  if.  .,    . 

smn#  dx  = + smn-*xdx 

J  n  .   n    J 

7r  .      ,.    .,     ,,      ,  cos  a;  sinn_1a; 

-  for  limits,  tJ 

2 

for  both  limits,  and  we  have 


we  take  o  and  -  for  limits,  the  term vanishes 

2  n 


IT  IT 

f*  .  fi  —  1  f*  . 

smn#ete  = siD.n~2xdx. 

Jo  w     Jo 


Now,  if  n  be  an  integer,  the  definite  integral  can  be 
easily  obtained ;  its  form,  however,  depends  on  whether  the 
index  n  is  even  or  odd. 

(1).  Suppose  the  index  even,  and  represented  by  2m, 
then 

It  IT 

J 2              2wi  —  i  rs* 
&\Vimxdx  = sm2m~zxdx. 
0                        2m    Jo 

Similarly, 

r^  2^  —  ■?  fs 

sin2wi-2^  = -     sin2m-4a?^; 

Jo  2m-2j0 

and  by  successive  application  of  the  formula,  we  get 
it 

f5"  •     o  ,  I    .    3    .    5  .  .    .   .    (2m-  i)      7T  ,    . 

8m2mxdx  = ^— -I 2 s .  -.        (5) 

Jo  2  .  4  .  6.  ...       2m         2         w/ 


120  Definite  Integrals. 

(2).  Suppose  the  index  odd,  and  represented  by  zm  +  1, 
then 

it  it 

fT  2m,     f* 

sin2m+1#dk  = sin2"*"1  #  da?. 

Jo  2m  +  ij0 

Hence,  it  is  easily  seen  that 

P«in"™*«fc  - 2.4.6..  2m   v  (6) 

Jo  3.5.7 (2m+i) 

Again,  it  is  evident  from  (31),  Art.  32,  that 


cosn  x  dx 


sinn#efc, 


and  consequently  (5)  and  (6)  hold  when  cos  x  is  substituted 
for  sin  x. 

it 

94.  Investigation  of      sinm#cosn#da>. 
Jo 

From  Art.  55,  when  m  and  n  are  positive,  we  have 

it  it 

f  2" .  w  —  1  P7  . 

smni#cosn#d#  =  &mmxcos>n-2xdx9 

Jo  m  +  n]o 

it  it 

and  ammxco8nxdx  = sinm-2a;cosn#^. 

Jo  m  +  n)0 

Hence,  when  one  of  the  indices  is  an  odd  integer,  the 
value  of  the  definite*  integral  is  easily  found. 


*  The  result  in  this  case  follows  also  immediately  from  Art.  92,  hy  making 
cos2  x  =  z ;  for  this  substitution  transforms  the  integral  into 

1  f1 


j    ( 1  -  z)m  z  2  dz. 


Elementary  Examples.  121 

For,  writing  2m  +  1  instead  of  m,  we  have 


m 


f2                                                                        2VYt  C% 

sin2m+1#  cosn#  e?#  = sin2"1"1^  cosn#  d#. 
0                                2«*  +  n  +  1 J  0 


Hence 

sin2m+1ffCOSn#<fo 

2m  (2m  -  2)  ....  2 
(2m  +  n+  i)(2m  +  n-  1)  .  .  .  .  ( 

2.4.6     ...       (2m) 


j; 


(»+  i)(»  +  3)  .  .  .  (w  +  2m  +  1)' 
In  like  manner, 


r     sin#  cosn#d# 

^+3)Jo 

(7) 


7T  7T 

T"2  2M   I  P^" 

sin2m#cos2n#d#  =  —. r     sin2w,#cos2n~2#d#. 

Jo  2(m  +  n)}0 


Hence 


n  ir    ■ 

2sin2m# cos2n# <fo  =  .!'3'^'"2W"\    2sin2m# dx 

Jo  (2W+2)  . .  .  (2m+2W)J0 

_   I  .  3  .  5  .  .  .  (2tt-  i)  .  I  .  3  .  5  .  .  .  (2m-  i)      7T  ,g. 

2.4.6 (2W+2W)'2>        V 

in  which  m  and  n  are  supposed  both  positive  integers. 

Many  elementary  definite  integrals  are  immediately  re- 
ducible to  one  or  other  of  the  preceding  forms. 

For  example,  on  making  x  =  tan  0,  we  get 

[**!_  -  LnM  =  '•3.5.--(*»-3)  .  *        (9) 

Jo(l+#2)n        Jo  2   .   4  .   6  .   .   .   (2»-2)       2  W/ 

Ca  - 

Similarly,  by  a?  =  a  sin  0,       #w  (a2  -  a2) 2  d#  transforms  into 

an+m+i  |2Sinn0cos»«+i0^. 


122  Definite  Integrals, 

fa  * 

{zax  -x*)2dx, 

e 

on  making  x  =  a  (i  -  cos  0),  becomes 

smm«  Odd. 


am+x  I  ah 


The  expressions  for  these  integrals,  when  m  and  n  are 
fractional  in  form,  will  be  given  in  a  subsequent  Artiole. 


Examples. 


f  2*  4* 

sin7  x  cos*  x  dx.  Ans.  - . 

Jo  3.5.7.  II 


*xdx. 


I   sin'j 

I 

[  (I  -  x2)ndx. 
J  0 

fi     a*" 

u77f 


5 .  10 .  20 .  30 .  40 

"    9- 19.  29.39-49* 

1.2.3...       (»*-*) 
"     w.  (»+i)  .  .  .  (n  +  m-  1)* 

2.4.6...      (2n) 

"      3.5.7..-    (2»+l)* 


1     a2»<fa;  i  .  3  .  5  .  .  .  (in  - 

— 7 •  »> 

•y^i-*2  2.4.6...       2» 


6. 


1   a.2n+l  ^  2  .  4.  .  6 

»» 


Jo^T 


a/i-*2  3.5  .7...  (2n+i) 

7.  Deduce  Wallis's  value  for  *  by  aid  of  the  two  preceding  definite  integrals. 
•      x»dx  A       2  .  4  .  6  ...(»- 1)        1 


8. 


.  Ans. , 

,(.+»^H  3.5.7....  »     a/"*"1 

when  n  is  an  ocfc*  integer. 
9.        I  ^{zax-xrfdx. 


Elementary  Examples.  123 

95.  Value  of  I    r*  xn  dx,  when  n  is  a  positive  integer. 

In  Art.  63  we  have  seen  that 


e~xxndx  =  -  e~xxn  + 


n    e~rxn~1dx. 


of1 
Again,  the  expression  —  vanishes  when  x  =  o,  and  also 

when  x  =  00  (Din2.  Calo.,  Art.  94,  Ex.  2). 

Hence  erxxndx  =  n      e~xxn-ldx.  (10) 

Consequently      \   e^x"  dx  =  1  .  2  .  3  . . .  n.  (11) 

Many  other  forms  are  immediately  reducible  to  the  pre- 
ceding definite  integral. 

For  example,  if  we  make  x  =  az  we  get 

jW*-1-^;--",  (.2) 

in  which  «  is  supposed  to  be  positive. 

Again,  to  find      %m  (log  x)n  dx ;  let  x  =  e"z,  and  the  in- 
tegral becomes 

(- 1)"  ["#***»**  =  (- 1)*  I;2'3;';n. 

•  Jo  (w+i)»+l 

Since  log  a?  =  -  log  ( -  ],  this  result  may  be  written  in  the 
form 


124  Definite  Integrals. 


The  definite  integral 


e~*afi*cb  is  sometimes  known  as 


The  Second*  Eulerian  Integral,  and  is  fundamental  in  the 
theory  of  definite  integrals.  Being  obviously  a  function 
of  n}  it  is  denoted  by  the  symbol  T[n)9  and  is  styled  the 
Gamma-Function. 

It  follows  from  (10)  that  '  l\A 

r{n+  i)  =  nr{n).  (14) 

Also,  when  n  is  an  integer  we  have 

r(n+  1)  =  1  .2.3  ..  .n.  (15) 

Again,  when  x  is  less  than  unity,  we  have 

=  1  +  x  +  x*  +  x*  +  &c ; 

1  -  x 

log#. =      loga?(i  +x  +  x*  +  ...)dx 

(by  a  well-known  result  in  Trigonometry). 
In  like  manner  we  get 


I 


1  log  x  dx         IT2 

,     I   +  X  12 


An  account  of  the  more  elementary  properties  of  Gamma- 
Functions  will  be  given  at  the  end  of  this  Chapter. 


*  The  integral  I   xm~l  (1  -  a;)w-1  dx,  considered  in  Art.  92,  is  sometimes  called 
Jo 
the  First  Eulerian  Integral ;  we  shall  show  suhsequently  how  it  can  he  ex- 
pressed in  terms  of  Gamma- Functions. 


Examples.  125 

Examples. 
I  |  log  f  -  ]  |   dx.  Ans.  i  .  2  .  3  ...  ft. 


■*xndx. 


i  .  2  .  .  .  n 


(log  a) 

IT2 


f1  ^  (log  a)2""1  ,  .  T         I  I  "1 

f1  dx  .        1 1  +  x\  ir2 

\    —  log  ( I .  Ans.  — . 

J0  x      b  \i  -xj  4 


96.  If  u  and  v  be  both  functions  of  x9  and  if  v  preserve  the 
same  sign  while  x  varies  from  x0  to  X,  then  we  shall  have 


rx  ex 

I     uvdx  »  U  I     vdx, 

J  *o  J x0 


where  TJ  is  some  quantity  comprised  between  the  greatest  and  the 
least  values  of  u,  between  the  assigned  limits. 

For,  let  A  and  B  be  the  greatest  and  the  least  values  of 
u,  and  we  shall  have,  when  v  is  positive, 

Av  >  uv  >  JBv ; 
when  v  is  negative, 

Av  <  uv  <  Bv. 

Consequently,  for  all  values  of  x  between  xQ  and  X  the 
expression  uvdx  lies  between  Av  dx  and  Bvdx,  and  accord- 
ingly, if  the  sign  of  v  does  not  change  between  the  limits, 

rx  t  rx  rx 

uvdx  lies  between  A  \    vdx  and  B       vdx, 

JXq  J  Xq  J  Xq 

which  establishes  the  theorem  proposed. 


126  Definite  Integrals, 

Cor.  If  f(x)  he  finite  and  continuous  for  all  values  of  a? 
between  the  finite  limits  x0  and  X,  then  the  integral 

r rw* 

will  also  have  a  finite  value. 

For,  let  A  be  the  greatest  value  of  f(x),  and  B  the  least, 

then       f{x)dx  evidently  lies  between  the  quantities 

rx  rx 

A       dx  and  B\    dx; 

Jx0  JxQ 

.•i       f{x) dx>B{X-  x0)  and  <  A  (X  -  x0). 

97.  Taylor's  Theorem. — The  method  of  definite  inte- 
gration combined  with  that  of  integration  by  parts  furnishes 
a  simple  proof  of  Taylor's  series. 

For,  if  in  the  equation 

f(X  +  h)-f(X)=^  f(x)dx 
we  assume  x  =  X  +  h  -  z,  we  get  dx  =  -  dz,  and  also 

CX+h  rh 

f{x)dx  =     f{X  +  h-z)dz; 
J  X  J0 

.-.  ,/(X  +  h)  -f{X)  =  P/'(X  +  h  -  z)dz. 
Again,  integrating  by  parts,  we  have 
\f\X  +  h-z)dz  =  zf{X  +  h-z)  +  \  zf\X  +  h-  z)dz. 
Hence,  substituting  the  limits,  we  have 

JV(X  +  h  -z)  dz  =  hf\X)  +  J"  zf(X  +h-z)  dz. 


Taylor's  Theorem.  127 

In  like  manner, 

\zf(X  +  h-  z)dz^f(X  +  h-z)  +^f"(X+h-z)dz, 
which  gives 

[hznX+h-z)d*  =  -f'(X)+[h-f'\X  +  h-s)d*; 

Jo  2  J o   2 


and  so  on. 

Accordingly,  we  have  finally 

ax+h)  =/(x) +i/(2) + JLr<n+.~.  ■ + jjrr/^'KJ) 


+ 


f  A  z*1"1  dz 

Jo/<.>(x+*-,)£*       (l6) 


This  is  Taylor's  well-known  expansion  * 

98.  Remainder  in  Taylor's  Theorem  expressed 
as  a  Definite  Integral. — Let  Bn  represent  the  remainder 
after  n  terms  in  Taylor's  series,  then  by  the  preceding  Article 
we  have 

/W(x+*-,)g  (I7) 

There  is  no  difficulty  in  deducing  Lagrange's  form  for 
the  remainder  from  this  result. 
For,  by  Art.  96,  we  have 

J  #I.  2. 3.. .(»-!)  i.2...n 

where  U  lies  between  the  greatest  and  least  values  which 
fW  (X  +  h  -  z)  assumes  while  z  varies  between  o  and  h. 


*  The  student  will  observe  that  it  is  essential  for  the  validity  of  this  proof 
(Art.  90),  that  the  successive  derived  functions,  /'(#), /'#'(?)»  &c.,  should  be 
finite  and  continuous  for  all  values  of  x  between  the  limits  X  and  X  +  h. 
Compare  Articles  54  and  75,  Dif.  Calc. 


128  Definite  Integrals. 

Hence,  as  in  Art.  75,  Diff.  Calc.  (since  any  value  of  z  between 
o  and  h  may  be  represented  by  (1  -  6)  h,  where  0  >  o  and  <  1 ) ; 
we  have 

Bn=  - /W(X+0h) 

1  .  2  . . .  n       v  ' 

where  0  is  some  quantity  between  the  limits  zero  and  unity. 

99.  Bernoulli's  Series. — If  we  apply  the  method  of 
integration  by  parts  to  the  expression  f(x)  dx  we  get 

\f(x)dx  =  xf[x)-[xf(x)dx; 

/.  Jy(s)  dx  =  Xf(X)  -/*/(*)  *<fe 
In  like  manner, 

and  so  on. 

Hence,  we  get  finally 

]"*/(«)*  =  f /(X)  -  ^/(X)  +  J—r/TX) -4c .-  (18) 

Compare  Art  66,  Diff.  Calc,  where  the  result  was  obtained 
directly  from  Taylor's  expansion. 

100.  Exceptional  Cases   in  Definite   Integrals. — 

In  the  foregoing  discussion  of  definite  integrals  we  have  sup- 
posed that  the  function  fix),  under  the  sign  of  integration, 
has  a  finite  value  for  all  values  of  x  between  the  limits.  We 
have  also  supposed  that  the  limits  are  finite.  "We  purpose  now 
to  give  a  short  discussion  of  the  exceptional  cases.*  They  may 


*  The  complete  investigation  of  definite  integrals  in  these  exceptional  cases 
is  due  to  Cauchy.  For  a  more  general  discussion  the  student  is  referred  to 
M.  Moigno's  Calcul  Integral,  as  also  to  those  of  M.  Serret  and  M.  Bertrand. 


Exceptional  Cases  in  Definite  Integrals.  129 

be  classed  as  follows: — (i).  When/(#)  becomes  infinite  at 
one  of  the  limits  of  integration.  (2).  "When /(a?)  becomes 
infinite  for  one  or  more  values  of  x  between  the  limits  of 
integration.  (3).  "When  one  or  both  of  the  limits  become 
infinite. 

In  these  cases,  the  integral      f(x)dx  may  still  have  a 


finite  value,  or  it  may  be  infinite,  or  indeterminate  :  depend- 
ing on  the  form  of  the  function  f(x)  in  each  particular  case. 
The  following  investigation  will  be  found  to  comprise  the 
cases  which  usually  arise. 

101.  Case  in  which /(a?)  becomes  infinite  at  one  of 
the  Limits. — Suppose  that  f(x)  is  finite  for  all  values  of  x 
between  x0  and  X,  but  that  it  becomes  infinite  when  x  =  X. 

The  case  that  most  commonly  arises  is  where  f{x)  is  of 

the  form  ■  .jl'         ,  in  which   \p(x)  is   finite  for   all  values 

(Jl  -  x) 

between  the  limits,  and  n  is  a  positive  index. 

Let  a  be  assumed  so  that  \p(x)  preserves  the  same  sign 
between  the  limits  a  and  X;  then 


(x   ^{x)dx   _  f' 


\jj{x)dx       [x  \fj{x)dx 


(X-x)n     Ja(X 


The  former  of  the  integrals  at  the  right-hand  side  is 
finite  by  Art.  96.  The  consideration  of  the  latter  resolves 
into  two  cases,  according  as  n  is  less  or  greater  than  unity. 

(1).  Let  n  <  1,  and  also  let  A  and  B  be  the  greatest 
and  least  values  of  \p(x)  between  the  limits  a  and  X  :  then, 
by  Art.  96,  the  integral 

t^M — r-  lies  between  A      7= r-  and  B     -7= r-, 

Moreover,  since  n  <  1,  we  have  evidently 

Cx     dx       _  (X  -  a)l-n 
Ja(X-*)»~      i-n    ' 

and  consequently,  in  this  case,  the  proposed  integral  has  a 
finite  value. 

[»] 


130  Definite  Integrals. 

(2).  Let  n  >  1,  and,  as  before,   suppose  A  and  B  the 
greatest  and  least  values  of  \p(x)   between  a  and  X;  then 


1: 


,  Jl  lies  between  A     y= r-  and  B     ,-==- :-. 

{X-x)n  )a{X-x)n  )a{X-x)n 


Again,  we  have 

fdx  i 

(X-x)n==  (n-  i)(J 


(X-x)n      (n-  i)(X-x)n-1 


Now  7= r — 7  becomes  infinite  when  x  =  X,  but  has  a 

(X  -  x)n~l  ' 

finite  value  when  x  =  a ;  consequently  the  definite  integral 

proposed  has  an  infinite  value  in  this  case. 

f      dx 

When  n  =  1,     -^= r  =  -  log  (X  -  x).     This  becomes 

J  (JC  —  x, 

infinite  when  x  =  X ;  and  consequently  in  this  case  also  the 

proposed  integral  becomes  infinite. 

The  investigation  when  f(x)  becomes  infinite  for  x  =  x0 
follows  from  the  preceding  by  interchanging  the  limits. 

102.  Case  where  f(x)  becomes  infinite  between 
the  Limits. — Suppose  f{x)  becomes  infinite  when  x  =  a, 
where  a  lies  between  the  limits  x0  and  X;  then  since 

f  f(x)  dx  m  \a  f{x)  dx+\  f  (x)  dx, 

the  investigation  is  reduced  to  two  integrals,  each  of  which 

may  be  treated  as  in  the  preceding  Article. 

\L(x) 
Hence,  if  we  suppose  f(x)  =  .         .n,  it  follows,  as  in 

the  last  Article,  that       f(x)dx  has  a  finite  or  an  infinite 

JXq 

value  according  as  n  is  less  or  not  less  than  unity. 

The  case  in  which. f(x)  becomes  infinite  for  two  or  more 
values  between  the  limits  is  treated,  in  a  similar  manner. 


Case  of  Infinite  Limits,  131 

For  example,  if 

f((h)  =  »,    /(«»)  =  oo,  .  .  .  /fa)  -  oo, 

where  au  a%  .  .  .  an  lie  between  the  limits  X  and  #0 ;  then 

cx  ca\  f°2  fz 

f(x)dx  =      f(x)dx+\    f(x)dx  +  &Q.  +  \     f(x)dx, 

}x0  J*0  *K  Ja» 

each  of  which  can  be  treated  separately. 

103.  Case  of  Infinite  Limits. — Suppose  the  superior 
limit  X  to  be  infinite,  and,  as  in  the  preceding  discussion,  let 

f{x)  be  of  the  form  .  *J  ;    ,  where  $(x)  is  finite  for  all  values 

of  x. 

As  before,  we  have 

f(x)dx=      f(x)dx  +  \  f(x)dx. 

The  integral  between  the  finite  limits  x0  and  a  has  a  finite 
value  as  before.  The  investigation  of  the  other  integral  con- 
sists again  of  two  cases. 

(1).  Let  n  >  t,  and  let  A  be  the  greatest  value  of  ^(x) 
between  the  limits  a  and  00,  then 


f    "dj(x\  dx  f 

,         .    is  less  than  A 
Ja(*-«)n  Jo 


dx 


(x  -  a)n 

[x     dx  1  1         _  1         1 

J  a  (x  -  af     n-i  L(a  -  a)""1      (X  -  a)-*  J 

The  latter  term  becomes  evanescent  when  X=  00  :  accord- 
ingly in  this  case  the  proposed  integral  has  a  finite  value. 

In  like  manner  it  is  easily  seen  that  if  n  be  not  greater  than 
unity,  the  definite  integral 

Cx     dx 

)a.{x-a)n 
T9a] 


132  Definite  Integrals. 

has  an  infinite  value  ;  and  consequently 

(""  \p(x)dx 

is  also  infinite,  provided  \p(x)  does  not  become  evanescent  for 
infinite  values  of  x. 

Hence,  the  definite  integral 

\p(x)dx 
Jx0  (x  -  a)n 

has,  in  general,  a  finite  or  an  infinite  value  according  as  n  is 
greater  or  not  greater  than  unity  :  \p(x)  being  supposed  finite, 
and  x0  being  greater  than  a. 

If  X become  -  oo,  a  similar  investigation  is  applicable;  for 
on  changing  x  into  -  x9  we  have 


f(x)dx  =  -\     f(-x)dx, 
Jx0  J-*0 


in  which  the  superior  limit  becomes  oo. 

104.  Principal  and  General  Values  of  a  Definite 
Integral. — We  shall  conclude  this  discussion  with  a  short 
account  of  Cauchy's*  method  of  investigation. 

Suppose  f{x)  to  be  infinite  when  x  =  a,  where  a  lies  be- 
tween the  limits  x0  and  X ;  then  the  integral  /  (x)  dx  is  re- 
garded  as  the  limit  towards  which  the  sum 


fa-fit  fZ 

f(x)dx+        f{x)dx 

JX0  Ja+vc 


approaches  when  e  becomes  evanescent ;  jm  and  v  being  any 
arbitrary  constants. 


*  This  and  the  four  following  Articles  have  been  taken,  with  some  modifica- 
tions, from  Moigno's  Calcul  Integral. 


Principal  Value  of  a  Definite  Integral.  133 

This  value  depends  on  the  nature  of/(#),  and  maybe 
finite  and  determinate,  or  infinite,  or  indeterminate. 

If  we  suppose  fi  =  v,  the  limiting  value  of  the  preceding 

sum  is  called  the  principal  value  of  the  proposed  integral ; 

while  that  given  above  is  called  its  general  value. 

ex    flx 

For  example,  let  us  consider  the  integral        — . 
Here  I*  *-fcmt  [[*-  +  H*! 

Also,  making  x  =  -  2, 

—  islogf—1;  while 

its  general  value  is  log  ( —  j  +  logf-j.     The  latter  expres- 
sion is  perfectly  arbitrary  and  indeterminate. 

('X    tfx 
-r. 
-^0  x 

-   £$-=-i— I 

.*.        -r  =  limit     — + — . 

J-*o  x2  Li«£      ve      X      xj 

Consequently,  both  the  principal  and  the  general  value  of  the 
integral  are  infinite  in  this  case. 


*  dx  _  1        1  < 

_a:0   «2         /U£        #0  ' 


134  Definite  Integrals. 

In  like  manner, 

Hence  the  general  value  of  the  integral  is  infinite,  while 
its  principal  value  is  -  (  —  -  —  \ 

It  may  be  observed  that  the  principal  value  of 

J_^- is  equal  to|^-. 

This  holds  also  whenever  f{x)  is  a  function  of  an  odd 
order:  i.e.  when/(-  x)  =  -/(#). 
For  we  have 

f0  f{x)dx=\X°f(x)dx  +  f°   /(*)<fo. 

J-*o  J°  J-*0 

But    r  /{»)  & = -  r  /( -  x)  dx  =  r °/( -  a  & ; 

J^o  J*o  Jo 

.-.  \*°  Ax)dx=\X°{f(x)  +f(-x))dx.  (19) 

J  -*0  Jo 

Accordingly,  if/(-  #)  =  -f(x),  we  get 

f*0 

■  f(x)  dx  =  o. 
J-*o 

Again,  if  fix)  be  of  an  even  order,  i.e.  if  /(-%)  =/(#)>  we 
have 

f*0  f*0 

/(#)  da?  =  2     /(#)  d#. 

105.  Singular  Definite  Integral. — The  difference 
between  the  general  and  the  principal  value  of  the  integral 
considered  at  the  commencement  of  the  preceding  Article  is 
represented  by 

Ca  +  fie 

f(x)  dx, 

Ja  +  ve 

in  which /(a)  =  00,  and  e  is  evanescent. 


Infinite  Limits.     Example.  135 

Such  an  integral  is  called  by  Cauchy  a  singular  definite 
integral,  in  which  the  limits  differ  by  an  infinitely  small 
quantity.  The  preceding  discussion  shows  that  such  an  in- 
tegral may  be  either  infinite  or  indeterminate. 

1 06.  Infinite  JLimits. — If  the  superior  limit  be  infinite, 

1 

regard      f(x)  dx  as  the  limit  of     f(x)  dx,  when  e  becomes 

J  #n  J  *0 


we 
evanescent. 


Also     f(x)  dx  =  limit  of        f(x)  dx  when  s  is  evanescent. 

In  the  latter  case  the  value  of  the  definite  integral  when 
fi  =  v  is,  as  before,  called  the  principal  value  of 


I    f(x)dx. 

In  this  we  assume  that  f(x)  does  not  become  infinite  for 
any  real  value  of  x. 

fix) 
107.  Example. — Suppose  '-^j\  *°^e  a  ra^onal  algebraic 

fraction,  in  which  f(x)  is  at  least  two  degrees  lowering  than 
F(x),  and  suppose  all  the  roots  of  F(x)  =  o  to  be  imaginary, 
it  is  required  to  find  the  value  of 

r  /M    , 

From  the  foregoing  conditions  it  follows  that  ==4  cannot 

l<[x) 

become  infinite  for  any  real  value  of  x :  accordingly  the  true 
value  of  the  integral  is  the  limit  of 


when  £  vanishes. 


136  Definite  Integrals, 

f(x\ 
To  find  this  value,  suppose  4nr4  decomposed  by  the  me- 
thod of  partial  fractions,  and  let 

A-B^/^l        .   A  +  By^l 

-=  and 


x  -  a  -  b^/-  i         x  -  a  +  b*/-  i 
he  the  fraotions  corresponding  to  the  pair  of  conjugate  roots 

a  +  by/-  i  and  a  -  b^/~  i,  of  F(x)  =  o ; 
then  the  corresponding  quadratic  fraction  is  the  sum  of 

A  -  By^i        .  A  +  By~i 
and 


x  -  a  -  b*/-  i         x  -  a  +  by/  -  i 

2A  (x  -  a)  +  2Bb 
1,e*      (x  -  a)z  +  b*     ' 

1 

•*•  I    T  : xi r?  =  2ttB  when  e  vanishes. 

}-—(x  -  a)2  +  b2 

1 

•'•  j  m±  (A.  _  ay  +  J»  "       °g  (v2  (1  +  ^E)2  +  6y6») 
=  2^4 log-,  when  e  =  o. 


f"    f(x) 
Investigation  of  \      'r-^—dx.  137 

i 

r™   {2A{x-a)  +  2B}dx  fu\ 

Hence ) '         ' —  =  2^  log  ^   +  2tt5.      (20) 

m« 

Now,  suppose  F(x)  to  be  of  the  degree  in  in  #,  and  let  the 
values  of  A  and  B,  corresponding  to  the  n  pairs  of  imaginary- 
roots,  be  denoted  by  Al9  A2,  .  .  .  Any  and  Bly  B%)  .  .  .  Bm  re- 
spectively ;  then  we  have 

1 

fte 

+  27r(i?1  +  i?2  +  •  •   •  +  i?n). 

Again,  since  f(x)  is  of  the  degree  2n  -  2  at  most,  we  have 
Ai  +  A2+  .  .  .  +  A„  =  o. 
For,  if  we  clear  the  equation 

f(x)      2  Ax  (x-ax)  +  2J#A  2  An  (x  -  an)  +  2Bnbn 


F  (x)  [x  -  atf  +  bx2  (x  -  anf  +  bj 

from  fractions,  the  coefficient  of  x2*1"1  at  the  right-hand  side  is 
evidently 

2  (Ax  +  A2  +  .  .  .  +  An) ; 

which  must  be  zero,  as  there  is  no  corresponding  term  on  the 
other  side. 

Accordingly  we  have,  in  this*  case, 

[.^)*"2'(2,,  +  *+,--+A)-         (2I) 


*  It  may  be  observed  that  when  f(x)  is  hut  one  degree  lower  than  F(x)y 

the  principal  value  of  I       "L.    dx  is  still  of  the  form  given  in  (21). 
J -»  *  [x) 


138  Definite  Integrals, 

"We  prooeed  to  apply  this  result  to  an  important  example, 
r  x2md 
Jo  i+% 


x2m  dx 
1 08.  Value  of  — —  when  m  and  n  are  Positive 


Integers,  and  n  >  m. 

Let  a  be  a  root  of  x2n  +  1  =  o,  and,  by  Art.  37,  we  have 


2WO'""1  2U 

Again,  by  the  theory  of  equations,  a  is  of  the  form 

iik  +  iW        • —   .    (2k  +  iW 

cos —  +  a/-  1  sin  s — , 

in  2n 

• 

in  which  k  is  either  zero  or  a  positive  integer  less  than  n ; 
.*.  a2m+1  =  cos  (2k  +1)6  +  a/^~i  sin  (2k  +  1)  0, 

where  0  =  . 

2n 

Hence  B  =  — —  ;  and  accordingly  we  have 

2n 

^!  +  B2  +  .  .  .  +  Bn  =  —  (sin  0  +  sin  30  +  . . .  +  sin  (2w  -  1)  0) . 

2W 

To  find  this  sum,  let 

8  =  sin  0  +  sin  30  +  .  .  .  +  sin  [m  -  1)  0 ; 
then 

2#sin  0  =  2  sin20  +  2  sin  0 sin  30  + . . .  +  2  sin  0  sin  (2»  -  1) 0 

=  1  -  cos  20  +  cos  20  -  cos  40  +  .  .  .  +  cos  [in  -  2)  0  -  cos  in9 

if 
=  1 -cos  in9=  2sin2w0  =  2sin2(2m+  1)  -  =  2  ; 


...s- 


sm0       .      2»»  +  i)tt 

sm — 

2n 


-?— Tdx.  139 

0  i  +  x2n 


Accordingly,  we  have 

r    x2mdx y 

J ,.  I    +  #2"  "  •      (2m+l)7T* 

J  w  sm — 

Hence,  by  (19), 

,a>  x?mdx       if"   a2OTflfo       tt  i 

0i  +  #2n  ~  2J_wi+a;2/*  "  2w    .    (2m  +  i)tt"  ^22' 


sm 

2W 


"We    now  proceed  to    consider  the    analogous  integral 

("  x2mdx 
-,  where  m  and  n,  as  before,  are  positive  integers, 
» 0 1  *"  x 
and  n  >  m.  ^ 

f*    #2ff* 
109.  Investigation  of     — - — -dx. 
Jo  *  ~  x 

"We  commence  by  showing  that 

fdx 

This  is  easily  seen  as  follows  : 

r"  e?#      r1  d#      r"  dx 

J0  1-0*  ~J0I  -«"      Jil  -#3' 
Now,  transform  the  latter  integral,  by  making  x  =  -,  and 
we  get 

Jll-Vjll-*"      Jol    ~^"      JoI-^; 

.*.       =  -  o. 


Again,  proceeding  to  the  integral 


r  x2m 
L 1  - 


1 40  Definite  Integrals, 

we  observe  that  i  +  x  and  i  -  x  are  the  only  real  factors  of 
i  -  x2n,  and  that  the  corresponding  partial  quadratio  fraction 
in  the  decomposition  of 

xim    .  I 

is 


i  -  x*n     n  ( i  -  x2) 

Consequently,  the  part  of  the  definite  integral  which  corre- 
sponds to  the  real  roots  disappears. 

Moreover,  it  is  easily  seen  that  the  method  of  Arts.  107 
and  108  applies  to  the  fractions  arising  from  the  n  -  1  pairs 
of  imaginary  roots,  and  accordingly 


x%mdx  ,_       _  „    . 
-  =  2tt{Bx  +  B2  +  .  .  .  +  ftuji 


where  Bl9  B2,  .  .  .  Bn.\  have  the  same  signification  as  before. 
Again,  since  the  roots  of  xZn  -  1  =  o  are  of  the  form 


kir         / —    .    kir 
cos  —  ±  a/-  1  sin  — , 
n  n 


it  follows,  as  in  Art.  108,  that 

Bx  +  Bt  + . . .  +  2?„_i  =  —  [sin  2d  +  sin  40  +  . . .  +  sin  2  (n  -  i)0], 

211 

_                                n      (2m  +  i)tt        ,    - 
where  0  = — ,  as  before. 


271 


Proceeding  as  in  the  former  case,  it  is  easily  seen  that 
sin  20  +  sin  40  +  ...  +  sin  2  (n  -  1)  0 

COS0  -cos  (2ft-  i)0         .  2m  +  1 

-  cot ir. 


2  sin  0  2n 


Hence 


x2mdx 

IT 

n 

cot 

2m  + 
2n 

I 

-„  1  -  x™ 

■  x2mdx 
ai-x*n 

IT 

2n 

cot 

2m  + 

2)1 

1 

-  7T. 

(23) 


Examples.  141 

Again,  if  we  transform  (22)  and  (23)  by  making  x~n  =  z 

_         2m  +  1  , 

and  a  = ,  we  get 

2U  ° 

Cza~ldz         7T  C*za-ldz  ,     v 

= ,  =  7TCOta7T.  [24] 

J0I+8       BinflJT  J  0  1    -   z 

The  conditions  imposed  on  m  and  w  require  that  a  should 
be  positive  and  less  than  unity. 

Moreover,  since  the  results  in  (24)  hold  for  all  integer 
values  of  m  and  n,  provided  n  >  my  we  assume,  by  the  law  of 
continuity,  that  they  hold  for  all  values  of  a,  so  long  as  it  is 
positive  and  less  than  unity. 

1 10.  The  definite  integrals  discussed  in  the  two  preceding 
Articles  admit  of  several  important  transformations,  of  which 
we  proceed  to  add  a  few. 

For  example,  on  making  u  =  za  in  (24),  we  get 


f"    du  a-n-  f"    du  , 

i  =  - ;  1  =  air  cot 

J  0 1  +  ua     sin  air  Jo  1  -ua 


aw. 


If  -  =  r,  these  become 
a 


fdu  ir  f"    du        IT      .  IT  .     ; 


rsm 
r 


where  r  is  positive  and  greater  than  unity. 
Again 

["  xndx  _  p  xndx       C  xndx 
Jo  1  +  &     Jo  1  +x2+  )xi  +zz' 

Now,  if  in  the  latter  integral  we  make  x  =  -,  we  get 

z 

p*  xndx  _  _  f°  z^dz  _  f1  x^dx 

J  ,  I   +  X2  "      J  ,  I   +  S2  " 


I 


"  #ne?# 


1  + 


1  +  ar 


dx.  (2.6) 


142  Definite  Integrals, 

Moreover,  from  (22),  when  n  is  less  than  unity,  we  have 

(27) 


x"dx 


I « i  +  x2              nw 
2  cos 

2 


Accordingly 


0    X  +  X~l     X  W7T 

2  cos  — 
2 


(28) 


In  like  manner,  it  is  easily  seen  that 

=  — tan— .  (29) 


'X  X*1  -  ST"  dx        7T  ,         M7T 


2 


It  should  he  noted,  that  in  these  results  n  must  he  less  than 
unity. 

Again,  transform  (28)  and  (29)  hy  making  x  =  e~irZ  and 
nw  =  a,  and  we  get 

r  eaz  +  e-az         1       a      C  eaz  -e~az  1      1  ,      a         ,     N 

-cfe=-sec-,         — -efe=-tan-         (30) 

J  0  enz  +  e**  2       2'    J  0  enz  -  e~*z  2        2 

We  add  a  few  examples  for  illustration. 

Examples. 

ra      dx  it 

J  0    ,  i#  '  .      7T* 

(a*  —  scn)n  wsin- 

dx 


f  dx 

Jo^  +  a^  +  i2? 


3- 


ti 


dx 


2ab  (a  +  b) 


IT 


& 


ta.nn9d9,  where  n  liea  between  +  I  and  -  i. 
Jo 


nir 

2C08  — 

2 


Differentiation  under  the  Sign  of  Integration.         143 

Clx"*+x'mdx     , 

I ,  where  n  >  m.  Ans. 

Jo  xn  +  xrn  x 


in  cos  — 
zn 

a        b 
2  cos  -  cos 


f«   (eax  +  e-ax\(^bx  _|_   e-bx)  2  2 

1     ax.  ,,  . 

Jo  en*  +  e~vx  cos  0  +  cos  b 

f 

Jo 


(e«T  +  p-ax^bx  _  e-bx\  ^  $ 

■ax. 


enx  _  g-n-x  cosa  +  cos  6 

It  should  be  observed,  that  in  these  we  must  have  a  -f  b  <  ir. 
8.  Hence,  when  b  <  ir,  prove  that 


I     cos  ax  ax  = 

Jo   eTT'  +  e-K* 


cos  axdx 

Jc 


o  ct*  -  em* 

ebx  4.   e-5x 


sin  ax  dx  = 


U*  +  *2Jcos^ 

£"  ■+-  2  COS  6  +  ^-« 

sin  b 

ea  +  2  cos  5  4-  £-"" 

I          c«  -  <r« 

f 

J  o   e*x  +  ^■'7rX  2  ea  +  2  cos  £  +  e~° 

Jo     I 


•1  ga  _  %-a  I 

<fc.  -4«s.  ir  cot  air . 


in.  Differentiation    of  Definite  Integrals. — It  is 

plain  from  Art.  86  that  the  method  of  differentiation  under 
the  sign  of  integration  applies  to  definite  as  well  as  to  in- 
definite  integrals,   provided   the  limits    of  integration   are 

independent  of  the  quantity  with  respect  to  which  we  dif- 
ferentiate. 

On  account  of  the  importance  of  this  principle  we  add  an 
independent  proof,  as  follows  : — 

Suppose  u  to  denote  the  definite  integral  in  question,  i.e. 
let 


u  = 


<p(v,  a)dx, 


where  a  and  b  are  independent  of  a. 

To  find  -j-  let  &u  denote  the  change  in  u  arising  from  the 
change  Aa  in  a  ;  then,  since  the  limits  are  unaltered, 


144  Definite  Integrals. 

Au  =       [<j>(x,  a  +  Act)  -  <p(x,  a)}dx; 

•    —  «  f  *  ^(^  a  + Aa)  -0(a?,  a) 
Aa      Ja  Aa 

Hence,  on  passing  to  the  limit,*  we  have 


du  _  Cb  dtp  (x,  a) 
da      L        da 


Also,  if  we  differentiate  n  times  in  succession,  we  ob- 
viously have 

d^u  =  f »  ^(a?,  q) 

tfa*      Ja         tfan 

The  importance  of  this  method  will  be  best  exhibited  by  a 
few  elementary  examples. 

112.  Integrals    deduced    by    Differentiation. — If 

the  equation 

p  OB 

e~ax  dx  =  - 
Jo  a 

be  differentiated  n  times  with  respect  to  a,  we  get 

JVr»«fa-  '-2air--, 

as  in  Art.  95. 

Again,  from  the  equation 

f "      dx      \    7T  1 

Jo  #2  +  a  ~  2  fll* 
we  get,  after  n  differentiations  with  respect  to  a, 
P"         db        _  7r  1  .  3  .  5  . ..  (in  -  1)    1 

Jo      (X*  +  fl)n+1  "22.4.6...  2tt       0^i; 

which  agrees  with  Art.  94. 

*  For  exceptions  to  this  general  result  the  student  is  referred  to  Bertrand's 
Calcul  Integral,  p.  1 8  j  . 


Differentiation  under  the  Sign  of  Integration.         145 

Again,  if  we  take  o  and  oo  for  limits  in  the  integrals  (23) 
and  (24)  of  Art.  21,  we  get 

Je"ax  cos  mx  dx  =  — =,  e"ax  sin  mx  dx  =  — ,.       (31) 

0  a?+m2     Jo  a2+  m2        v     ' 

Now,  differentiate  each  of  these  n  times  with  respect  to  a> 
and  we  get 

fax  xn  cos  mx  fa  m  t      j  \n  f       \    (        g       \ 

Jo  \daj  \a2  +  my 

|_w.cos(w  +  i)0 


(a2  +  m2)' 


Ti   » 


f       —  «  •  ^         »  .  sin  (w+  i)0 

0  n+_l     >  U2/ 

(a2  +  m2)  ^ 

where  w  =  a  tan  0.     (See  Ex.  17,  18,  Diff.  Calc,  pp.  58,  59.) 
Next,  from  (24)  we  have 


r  x^_dx  _ 

Jo       1~X~ 


COt  «7T. 


Accordingly,  if  we  differentiate  with  respect  to  a,  we  have 

( °°  af^1  log  x  dx  _         7T2 
Jo         1  -  x  sin2 air' 

Again,  if  the  equation 


JV*-i 


be  transformed,  by  making  y  = 7-,  it  evidently  gives 

a  +  ox 

x71^  dx  1 


r     of-1 

Jo     (a  +  i 


[10] 


146  Definite  Integrals. 

Now,  differentiating  with  respect  to  a,  we  have 

J0    (a  +  bx)M2  ~  n(n  +  i)a26»' 

If  we  proceed  to  differentiate  m  -  i  times  with  regard  to 
a,  we  have 


f"      x" 

lo    (a  + 


af'^dx  i .  2  .  3  . . .  (m-i) 


o    (a  +  bx)m+n     n.(n+  i)(n  +  2)  . . .  (n  +  m  -  1)  '  am6n* 

113.  By  aid  of  the  preceding  method  the  determination 
of  a  definite  integral  can  often  be  reduced  to  a  known  integral. 
We  shall  illustrate  this  statement  by  one  or  two  examples. 
Ex.  1.  To  find 


j: 


log(i  +  sin  a  cos  a*) 

~— ~ -— ■ ~~~ ■""■""■ — — ~~~^^^—  ax. 


cos# 


Denote  the  definite  integral  by  u,  and  differentiate  with 
respect  to  a  ;  then 


du 
da 


f*       cosaefo  .,       .    ,      rt. 

=      : =  7T  (by  Art.  18). 

J  0  1  +  sm  a  cos  x         s  ' 


Hence,  we  get 

dxlog(i  +  sin  a  cos  a? 


I? 


=  7ra. 

COS# 


No  constant  is  added  since  the  integral  evidently  vanishes 
along  with  a. 


-™  e~ax  sm  mx  7 


■■J. 


In  this  case 

a 


e"ax  cos  mx  dx 


dm     Jo  a2  +  m? 

f    dm         .        fm\ 

.\  u  =  0    - -  tan"1   —  . 

Ja2  +  m2  \aj 

No  constant  is  added  since  u  vanishes  with  m. 


Case  where  the  Limits  are  Variable.  147 

Ex.  3.  Next  suppose 

4-  n?^ 

dx. 


-. 


log(i  +  #V) 


Here 


da 


1  +  &V 
2ax2dx 


0    (1  +  a2x*)(i  +  b2x2) 

1      r  r     2adx    _  r     zadx  "I 
J0    1+6V     J0    i+aVj 


-62 


I  /fl  \  7T  ( 


_w  f  da 
U  ~b}aT 


-  =  I"  log  (a  +  J)  +  const. 


To  determine  the  constant :  let  a  =  o,  and  we  obviously 
have  u  =  o. 

Consequently,  the  constant  is  -  ■?  log  b; 

The  method  adopted  in  this  Article  is  plainly  equivalent 
to  a  process  of  integration  under  the  sign  of  integration. 
Before  proceeding  to  this  method  we  shall  consider  the  case 
of  differentiation  when  the  limits  a  and  b  are  functions  of 
the  quantity  with  respect  to  which  we  differentiate. 

114.  Differentiation  where  the  Limits  are  Va- 
riable.— Let  the  indefinite  integral  of  the  expression 
<p(x,  a)dx  be  denoted  by  F(x,  a) ;  then,  by  Art.  91,  we  have 

rb 

u  =      <j>(xt  a)  dx  =  F(b,  a)  -  F(a,  a); 
du     d .  F(b,  a)      , ,.     . 
[10  a] 


148  Definite  Integrals. 

and  —  = - =  -d>{aia). 

dx  da  r\  >    / 

Again,  taking  the  total  differential  coefficient  of  u  re- 
garding a  and  b  as  functions  of  a,  we  have 

du      fb  d<j>(x,  a)  du  db      du  da 

da     ja      du  db  da      da  da 

1*  d<b(x.  a)  _  ..     sdb        .       .da    ,     . 

,^*+*<*.  «)£-♦(*«)£•  (33) 

By  repeating  this  process,  the  values  of  — ,  — ,  &c,  can 

be  obtained,  if  required. 

115.  Integration  under  tbe  Sign  of  Integration. — 

Eeturning  to  the  equation 


u  =      <f>(x,  a)dx, 


where  the  limits  are  independent  of  a,  it  is  obvious,  as  in 
Art.  87,  that 

uda  =  <i>(®,  a) da    dx, 

provided  a  be  taken  between  the  same  limits  in  both  cases. 
If  we  denote  the  limits  of  a  by  a0  and  al}  we  get 

uda  =  <J>(%,  a)  da    dx, 

Ja0  Ja\_Ja0  J 

(p(x,  a)dx  \da=  <p(x,  a) da  \dx.       (34) 


or 


This  result  is  easily  written  in  the  form 

'ax  ft 


<j>(xt  a)dxda=\         <j>(x,  a)dadx,  (35) 

J  a0  Ja  Ja  J  a0 


Integration  under  the  Sign  of  Integration.  149 

These  expressions  are  called  double  definite  integrals,  as  in- 
volving successive  integrations  with  respect  to  two  variables, 
taken  between  limits. 

It  may  be  observed  that  the  expression 


'da 
is  here  taken  as  an  abbreviation  of 


<j>(x,  a)dX( 

J  aa  Ja 


.c 


(j)(x,  a)dx 


da, 


in  which  the  definite  integral  between  the  brackets  is  sup- 
posed to  be  first  determined,  and  the  result  afterwards 
integrated  with  respect  to  a,  between  the  limits  a0  and  ax. 
The  principle*  established  above  may  be  otherwise  stated, 
thus  :  In  the  determination  of  the  integral  of  the  expression 

<j>(x,  a)dxda 

between  the  respective  limits  x0i  xx,  and  a0,  ax,  we  may  effect  the 
integrations  in  either  order,  provided  the  limits  of  x  and  a  are 
independent  of  each  other. 

In  a  subsequent  chapter  the  geometrical  interpretation  of 
this,  as  well  as  of  a  more  general  theorem,  will  be  given. 

We  now  proceed  to  illustrate  the  importance  of  this 
method  by  a  few  examples. 

1 1 6.  Applications  of  Integration  under  the  Sign  J. 

Ex.  i .  From  the  equation 


we  get 


i 


1      ,  J        i 
x*'1  dx  =  - 


££*•*-  C*-*® 


*  It  should  be  noted  that  this  principle  fails  whenever  4>(a?,  a),  or  either  of 
its  integrals  with  respect  to  o,  or  to  x,  becomes  infinite  for  any  values  of  x  and  a 
contained  between  the  limits  of  integration.  The  student  will  find  that  the 
examples  here  given  are  exempt  from  such  failure. 


150  Definite  Integrals, 

Henoe 

Again,  if  we  make  x  -  e~z  in  this  equation,  we  get 

£  =**£*& 

Ex.  2.  "We  have  already  seen  that 

Je-"*  cos  m#cfo;  =  -= - 
o                               a3  +  m% 

Henoe 

feudal  cos  m#d#  =         „a  a  = 
0      JLa0  J  Ja0«2  +  ^2 

I .      (a?  +  m2\ 

or  cosmxdx  =  -  log   — r =  . 

Jo  •  2     &W  +  m2J 

Ex.  3.  Again,  from  the  equation 

/.OS 

er0*  sin  »Mrdb  =  -= -, 

J0  a*  +  m29 

we  get 

Jo   J«0  Jaoa2  +  m2' 

sin  mxdx  =  tan-1  (  — )  -  tan"1  [  —  \ 

Jo  x  \mj  \mf 

Compare  Ex.  2,  Art.  113. 

If  we  make  ao  =  o  and  ax  =  00  in  the  latter  result,  we 
obtain 

J     sin  ma?  ,       7T 
dx  =  -. 
0        x  2 


151 

Value  of  |   e~x2dx. 

0 


Ex.  4.  To  find  the  value  of 

f  e~xadx. 

Denoting  the  proposed  integral  by  k}  and  substituting 
ax  for  xy  we  obviously  have 

[  e-**x°adx  =  k; 
.-.  f  e'^^adx  =  ke-°-\ 

[  e°-*^x*)adadx=k  [  e~a*da  -  7c2. 

J0  2I+08 

2j0I+»2  4 

Hence  f'*' 

\  tr^dx-k^-^/H.  (36) 

Jo  2 

This  definite  integral  is  of  considerable  importance,  and 
several  others  are  readily  deduced  from  it. 
117.  For  example,  to  find 


Hence 


But 


-X* 

Z2dx. 


Here 


(A)      •-£ 

du  _        f"  -«a-ji    dx 
da  Jo  x2' 


Again,  let  s  =  -,  and  we  get 
x 


152  Definite  Integrals. 

.'.  —  =  -  2M  ;  hence  w  =  Ce*2a. 
da 

To  determine  (7,  let  a  =  o,  and,  by  the  preceding  example, 

i#  becomes  ^— . 
2 

Consequently 

]/      -^=Ye"2a-  (37) 

Again,  to  find 

(B)       m  -  j  <r°9*9cos  ibxdx. 


Here 


—  =  -  2  I  e""0*3©^  2&i?#dk. 


But,  integrating  by  parts,  we  have 


f      n*r*    •        t         7  e  BiIL2bx        2b  f    _-,  ,      _ 

2  \  e~a  x  sm  2bxxdx  = = +  —    e-°  *  cos  2bxdx: 

J  a2  a2  J 

.*.      e"°2a;asin  2bxxdx  =  -r     e-03*3  cos  2&rflfc. 
Jo  a2  J o 


Heroe 

du  _     2bu        du  _      2bdb 
db~~~tf'°T^"=        "*"* 

Hence  u  =  Ce  a*; 


Also,  when  b  =  o,  u  becomes  — — ; 

2a 


.'.  [  e-"'*' cob  2bxdx  =  YLe~a\  (38) 


Examples. 


153 


Again,  if  we  differentiate  n  times,  with  respect  to  a,  the 
equation 


t 


e~aX*dx 


2>/a 

and  afterwards  make  a  =  i,  we  get 

i  .  z  .  s .. .  (m-  i) 


(C) 
Next,  to  find 

(0) 

We  obviously  have 


/" 


cos  mxdx 


C  cosm 
Jo    i  + 


ae-ft3M4  = 


i  +  x 


i  > 


-j: 


a  e~a2  (1+a;2)  cos  mx  dx  d< 
But,  by  (38),  we  have 

2     e^***  cos  mxdx 

.'.  */v      e~aZ~wda 


f   cos  mxdi 

a  =       r 

Jo     1  +  sr 

=  - —  e  ^ 
a 

)m  cos  m, 
0     1  + 


cos  mxdx 

,2   ■ 


Hence,  by  (37),  we  have 


fcos 
0  ~i 


cos  mxdx 


+  X"  2 

Again,  differentiating  with  respect  to  m,  we  obtain 
'sinmxdx      ir    _ 


#sn 
Jo      1 


+  af 


e  '". 


(39) 


(40) 


154 

Definite  Integrals. 

Examples. 

i. 

Jo  «*»  -  #■*• 

Ans. 

-sec2-. 
4        * 

2. 

P  l  a;*"1  +  a;-*  rf# 
J2     I  +«      a;" 

>» 

log  (tan  ^), 

when  i 

1  >  o  and  <  i. 

3- 

f  i  z*  +  z~a  -  2   dz 
Jo        i  -  *      log  z 

» 

logf^l 
\sin  o7T/ 

4- 

1  2  log  (i  +  cos  6  cos x) 
Jo                                   < 

dm 

;os  a;' 

>> 

H?-)- 

5- 

joC°"10gU  +  a») 

dx. 

>> 

ir  (<r«  -e-pj. 

6. 

f"  zr  log  zdz 
Jo      i  +  z2   * 

n 

.   rir 

sin  — 

fl-2            2 

4       or* 
*  cos2 — 

2 

P"  sinaflifl 
Jo  enO—e-irO' 

I  *<*-   I 

7- 

>> 

4<?a+   I* 

1 1 8.  The  values  of  some  important  definite  integrals  can 
be  easily  deduced  from  formula  (31),  Art.  32. 
For  example,*  to  find 

[2  log  (sin  B)dB. 

IT  IT 

Here  f  *  log  (sin  0)  dB  =  I  log  (cos  0)  dB. 

Hence,  denoting  either  integral  by  u,  we  have 

zu  =    2  {log  (sin  B)  +  log  (cos  B) )  dB 
1 A 


*  These  examples  are  taken  from  a  Paper,  signed  "H.  G.,"  in  the  Cambridge 
Mathematical  Journal,  Vol.  3. 


Theorem  of  Frullani. 

ir 
f 2  7T 

=      log  (sin  2&)dQ  —  log  2. 
Jo  2 

Again,  if  2  =  20,  we  have 

ir 

log  (sin  20)e?0  =  -      log  (sin  z)dz 
0  2J  0 

=  -      log  (sin  z)c?s  +  -      log  (sin  s)cfe  ; 

2  J    0  2   J    7T 

2 
but,  since  sin  (n  -  z)  =  sin  2, 

log  (sin  s)cfe  =      log  (sin  s)  cfe. 
2" 
Consequently 

ir  ir 

['log (sin 20)d0  =  ['log  (sin 0)48 ; 


155 


.-.      Iog(sin0)tf0  =  --log(2).  (41) 

Jo  2 

Again,  to  find 

[*0  log  (sin  0)d0. 

Here 

[*  0  log  (sin  0)  dO  =  ["  (ir  -  0)  log  (sin  0)^0 ; 

.-.  [*  01og  (sin  9)49  =  -  flog  (sin  0)^0  =  -  -  log  (2). 

Jo  2 J  0  2 

119.  Theorem  of  Frullani. — To  prove  that 


rK„~,  ^  yv,~;  ^  _  ^^  ^ 


$(a#)  -  0(for) 


156  Definite  Integrals, 

Let  u  =     — 21Li  ^2  j  substitute  ax  for  s,  and  we  get 

Jo  2 

h 

Jo  X 

If  we  substitute  b  for  «,  we  get 


'^W-^o)^ 


h 


Jo  X 


ft 


p<b(ax)dx      [b(h(bx)dx       ,  x  Udx        .  N,      I     ,     N 

j„     *     "Jo     *     -*(°)],7-*<0)1<«5-  (42) 


Hence 


"<b(ax)-6(bx)  ,      fid>(bx)dx        .  t_      /ft\         ,     N 

..    ,   **-)»« — »(°M«)  (43) 

a 

If  we  suppose  h  =  oo,  we  get 

J0     ; *-.#(€)  fcgjyt  (44) 

A 

provided  £LJ  ^  =  o  when  ^  =  oo. 

\h     x 

b 

For  example,  let  (j>(x)  =  cos  a?,  and,  since  the  integral 


1 

'6  cos  bx  _ 

h      x 


evidently  vanishes  when  h  =  oc,  we  have 

f"  cos  ax  -  cos  bx  ,       ,      J 

d#  =  log  -. 

Jo  x  3  a 


Theorem  of  Frullani.  157 

Frullani's  theorem  plainly  fails  when  <j>(ax)  tends  to  a 
definite  limit  when  x  becomes  infinitely  great.  The  formulae 
can  be  exhibited,  however,  in  this  case  in  a  simple  shape,  as 
was  shown  by  Mr.  E.  B.  Elliott* 

For,  in  (42)  let  h  =  ab,  and  it  becomes 

[h  <b(ax)dx      Ca  <b(bx)dx        ,  *  ,     /5\  1     \ 

J.VHo  *  =*(o)l0gt>     (45) 

Again,  if  0(  00)  denote  the  definite  value  to  which  $(ax) 
tends  when  x  increases  indefinitely,  then  when  h  becomes 
infinite  we  may  substitute  $(oo)  instead  of  <t>{bx)  in  the 
integral 


h 

6  <j)(bx) 


dx\ 


in  which  case  it  becomes 


7=*(«>)log(f} 


On  making  this  substitution  in  (43),  we  get 

J#"^^*-j*(-)-#(o)jfcg(;}   (46) 

For  example,  let  <p(ax)  =  tsnrl(ax)  then  we  have  0(o)  =  o, 
and  ^(oo)  =  -. 

Accordingly  we  have 


tan-1  ax  -  tan 


2  )h  x      2     °\bj 


*  Educational  Times,  1875.  The  student  will  find  some  remarkable  exten- 
sions of  the  formulae,  given  above,  to  Multiple  Definite  Integrals,  by  Mr.  Elliott, 
in  the  Proceedings  of  the  London  Mathematical  Society,  1876,  1877.  Also  by 
Mr.  Lendesdorf,  in  the  same  Journal,  1878. 


158  Definite  Integrals. 

119a.  Remainder  in  Lagrange's  Series. — We  next 
proceed  to  show  that  the  remainder  in  Lagrange's  series 
(Diff.  Calc,  Art.  125)  admits  of  being  represented  by  a 
definite  integral.  This  result,  I  believe,  was  first  given  by 
M.  Popoff  (Comptes  Rendus,  1861,  pp.  795-8). 

The  following  proof,  which  at  the  same  time  affords  a 
demonstration  of  the  series,  of  a  simple  character,  is  due  to 
M.  Zolotareff : — 

Let  z  =  x  +  y  <f>(z)  ;  and  consider  the  definite  integral 


-     {y<j>{u)  +x-  u}nF\u)du. 


Differentiating  this  with  respect  to  x,  we  get,  by  (33), 
Art.  114, 

5£-»«m  -**(♦(*)}•  W  (47) 

If  in  this  we  make  n  =  1 ,  we  get 

but  s0  =  F{z)  -  F(x)  ; 

.:F(z)=F(x)+y<p(x)F'(x)  +  ^.  (48) 

In  like  manner,  making  n  =  2,  we  have 


"1 -trim*  p® +  ■%: 


d2s2 


Substituting  in  (48)  it  becomes 
Again, 


Gamma  Functions. 


159 


i    dzs2 
1.2  dx2 


I  .  2 


$dx2 


(♦(ij)'^W] 


i      6?3g3 
1.2.3  ^3 ' 


wsn-i  =  ^n{^)  )"*>)  + 


<fe' 


dn-lsn.x 


dn- 


1  .  2 


n  - 1    e?#* 


1  .  2  . . .  ft  efa?w" 


{*(*))"*»] 


r/J 


1.2  . . .  n  dxn 


Hence  we  get  finally 


+  &o 


_J__(± 
1.2...  w  \db 


[y  #(*)  +  *  "  if\nF'(u)  du.        (49) 


Consequently  the  remainder  in  Lagrange's  series  is  always 
represented  by  a  definite  integral. 

We  next  proceed  to  consider  a  general  class  of  Definite 
Integrals  first  introduced  into  analysis  by  Euler. 

120.  (Mamma  Functions. — It  may  be  observed  that 
there  is  no  branch  of  analysis  which  has  occupied  the  atten- 
tion of  mathematicians  more  than  that  which  treats  of 
Definite  Integrals,  both  single  and  multiple ;  nor  in  which 
the  results  arrived  at  are  of  greater  elegance  and  interest. 
It  would  be  manifestly  impossible  in  the  limits  of  an 
elementary  treatise  to  give  more  than  a  sketch  of  the  results 
arrived  at.  At  the  same  time  the  Gramma  or  Eulerian 
Integrals  hold  so  fundamental  a  place,  that  no  treatise, 
however  elementary,  would  be  complete  without  giving  at 
least  an  outline  of  their  properties.  "With  such  an  outline 
we  propose  to  conclude  this  Chapter. 

The  definitions  of  the  Eulerian  Integrals,  both  First  and 
Second,  have  been  given  already  in  Art.  95. 

The  First  Eulerian  Integral,  viz., 

aP-^i  -x)n-ldx, 

is  evidently  a  function  of  its  two  parameters,  m  and  n ;  it  is 
usually  represented  by  the  notation  B(my  n). 


160  Definite  Integrals. 

Thus,  we  have  by  definition 

f1  of*-1  (i  -  x)n-ldx  =  B(m,  n). 

[  (50) 

e~xxp-1dx  =  T(p). 

The  constants  m,  n,  are  supposed  positive  in  all  cases. 
It  is  evident  that  the  result  in  equation  (14),  Art.  95,  still 
holds  when  p  is  of  fractional  form. 
Hence,  we  have  in  all  cases 

r(p  +  i)  =  pT{p).  (51) 

This  may  be  regarded  as  the  fundamental  property  of 
Gamma  Functions,  and  by  aid  of  it  the  calculations  of  all 
such  functions  can  be  reduced  to  those  for  which  the  para- 
meter p  is  comprised  between  any  two  consecutive  integers. 
For  this  purpose  the  values  of  T{p)y  or  rather  of  log  V{p\ 
have  been  tabulated  by  Legendre*  to  1 2  decimal  places,  for 
all  values  of  p  (between  1  and  2)  to  3  decimal  places.  The 
student  will  find  Tables  to  6  decimal  places  at  the  end  of  this 
chapter.  By  aid  of  such  Tables  we  can  readily  calculate  the 
approximate  values  of  all  definite  integrals  which  are  re- 
ducible to  Gamma  Functions. 

It  may  be  remarked  that  we  have 

r(i)  =  i,     r(o)=oo,     r(-jp)=oo, 

p  being  any  integer.     For  negative  values  of  p  which  are 
not  integer  the  function  has  a  finite  value. 

Again,  if  we  substitute  zx  instead  of  a,  where  2  is  a  con- 
stant with  respect  to  x,  we  obviously  have 

P '.f-ap«*;£M1  (52) 


*  See  Traite  des  Fonctions  Elliptiqim,  Tome  2,  Int.  Euler,  chap.  1 6. 


Expression  for  B(m,  n). 


161 


With  respect  to  the  First  Eulerian  Integral,  we  have 
already  seen  (Art.  92)  that 


3^(1  -  x)n^dx  =-• 


xn~l  (1  -  x)m~xdx\ 


.'.  B  (m,  n)  =  B(n,  m). 

Hence,  the  interchange  of  the  constants  m  and  n  does  not 
alter  the  value  of  the  integral. 

Again,  if  we  substitute for  x,  we  get 


1  +  y 


[  xm~Hi  -x)n~ldx  =  f     J£ 

Jo  Jo    (1  + 


dy 


ft 


f        ym   dy 

Hence  -. ^—  =  B(m.  n). 

Jo    (1  +y)m+n        K   '    ; 


(53) 


We  now  proceed  to  express  B  (m,  n)  in  terms  of  Gamma 
Functions. 

121.  To  prove  that 

B [m,  n)  =   „ ,  ' — v. 

From  equation  (52)  we  have 

r(f»)>       e-zx%mxm~xdx. 


Hence 


T  (m)  e~z  sM_1  - 

,*.  T(m)  I     e-zzn-1dz  =  \ 
J  0  Jo 


M 


a^^dx. 


162  Definite  Integrals. 

But,  if  2  (i  +  x)  =  y,  we  get 

J0  (i  +x)m+n)o  (i   +#)m+n 

.-.  r(m)  r(«)  =  r(m  +  n)  [  (i^^ln. 


»)• 


Accordingly,  by  (53),  we  have 


„,      .    r(w)  r(n)  ,    1 

B(m,  n)  =  — V —  \  ■  (54) 

v        '       r(m  +  n) 


(55) 


Again,  if  m  =  1  -  n,  we  get,  by  (24), 

r(»)  r(i  -  n)  =      =  -. —  . 

Jo     1  +x      sinw7r 
If  in  this  n  =  -,  we  get 

This  agrees  with  (36),  for  if  we  make  #2  =  2,  we  get 

f     er^dx  =  -\    e~*  s-4  eft  =  -A24  (56) 

Again,  if  we  suppose  in  the  double  integral 

^y^dxdy 

x  and  y  extended  to  all  positive  values,  subject  to  the  condi- 
tion that  x  +  y  is  not  greater  than  unity ;  then,  integrating 
with  respect  to  y,  between  the  limits  o  and  1  -  x,  the 
integral  becomes 

»J.*     (       X)  dX~  n    r(m  +  n+i)  '^«4), 

xm~l yn~x dxdy  =     .v    ; — *-^- ;  (57) 

in  which  x  and  y  are  always  positive,  and  subject  to  the  con- 
dition x  +  y  <  1 . 


Gamma  Functions.  163 

122.  By  aid  of  the  relation  in  (54)  a  number  of  definite 
integrals  are  reducible  to  Gamma  Functions. 
For  instance,  we  have 

f  ym~ldy       f1    ym~ldy    +  f   ym'ldy 
Jo  (1  +  y)m*n  "  J  0  (1  +  y)m+n  +  J 1  (1  +  2/)m+"' 

Now,  substituting-  for  y  in  the  last  integral,  we  get 

x 

r    ym~xdy         f1     x^dx 

Ji(i  +  ^)w+n=Jo(i+^r+"' 

Hence 

fl^  +  ^  =  rwr(«) 


0(i+^)m+n  T(m  +  w) 


Next,  if  we  make  *  ■*  -p  we  ge^ 

rj**d*_  ['    ym-ldy    . 

Jo(i+*)—"        Jo(«y  +  J)-H,f 

r     y"*-1^  r(m)  V(n) 

""'  Jo  («y  +  &)m+n  ~  ambnr{m  +  n)' 
Again,*  let  #  -  sin2  0,  and  we  get 

['^(i  -a)""1^  =  2  P  sin^Gcos^fldfl; 


This  result  may  also  be  written  as  follows  : 
'sin?-'  61  cos'-  9d8=     v7     W. 

Jo  2^ 


(59) 


W-ecos-9^=lMi>)  (60) 

0  2  r(m  +  w) 


(61) 


*  These  results  may  be  regarded  as  generalizations  of  the  formulae  given  in 
Arts.  93,  94,  to  which,  the  student  can  readily  see  that  they  are  reducible  when 
the  indices  are  integers. 


164  Definite  Integrals. 

If  we  make  q  =  i,  we  get 


psinP"1  ddO  =  ^     /2'    v.  (62) 

Jo  2        /p  +  i\ 

\ 2  y 


Again,  if  ^?  =  q  in  (61)  it  becomes 


LAM.  =  I  \mP-lBco^6dd  =  -4t  h 

2TCp)        Jo  2*-lJo 

Let  20  =  2,  and  we  have 

8  sin*"1 2  0  dO  =  -     sinP"1  zdz  =  |  sin*-1  2  afe 
Jo  2J0  Jo 

^% 

He.ee         r(f)r(*±i)  -  £r«. 

If  we  substitute  2  m  for^?,  this  becomes 


r(»)r(»  +  ij-^r(2»).  (63) 

Again,  make  y  =  tan20  in  (59),  and  we  get 

T    sin^flcos^fl^fl  r(m)  T{n) 

0  (a  sin20  +  b cos20)m+ft  ~  iam  bnT(m  +  w)*  '  4' 

123.  To  find  the  Value*  of 

n  being  any  integer. 

*  This  important  theorem  is  due  to  Euler,  hy  whom,  as  already  noticed,  the 
Gamma  Functions  were  first  investigated. 


^o/rQrgjr©...^ 


165 


Multiply  the  expression  by  itself,  reversing  the  order  of 
the  factors,  and  we  get  its  square  under  the  form 

that  is,  by  (55), 


.7r.27r.37r        .    (n  -  i)tt 
sin  -  sin  —  sin  — ...  sin 

n         n  n  n 

To  calculate  this  expression,  we  have  by  the  theory  of 
equations 

1  -x2n 

1  -x2 

(       «■  *v        2*  a  f        (»-*)*  2^ 

=(  1  -2#cos-+#2  If  1  -2#cos — +<r  1...I  1  -2#cos- — +ar  1. 

Making  successively  in  this,  #  =  1,  and  a?  ■  -  1,  and  re- 
placing the  first  member  by  its  true  value  w,  we  get 

/      .     7T  \y      .     27rV        f      •     (n--  QttV 

w  =    2  sin  —     2  sin  —    ...    2sin —    , 

\  2»/  \  2nJ         \  2n      J 

(  7r\y  2tt\2        /  (n-  i)7rV 

ft  =      2   COS  2    COS  ...      2   COS  , 

\  2)\)  \  2UJ  \  2U        J 

whence,  multiplying  and  extracting  the  square  root, 

„  ,  •    w    .     27r  .    (n  -  i)ir 

n  =  2"-1sm  —  sm  —  ...  sin — . 

n         n  n 

Hence,  it  follows  that 


166  Definite  Integrals, 

124.  To  find  the  values  of 

€*x  cos  bxxm~l  dx,  and       e-°x  sin  bx  x™-1  dx. 

If  in  (52)  a  -  b*f-  1  be  substituted*  for  z  the  equation 
becomes 

,        f"  e-a*e^^dx  m  r(m)_     m  r(m)(a  +  b^T)^ 

{a-b*/-\)m  (a2  +  b2)m 

Let  a  =  (a2  +  &2)*  cos  0,  then  5  -  (a2  +  62)*  sin  0,  and  the 
preceding  result  becomes 

^'(cos  bx  +  »/-  1  sin  bx)xm~ldx 
=     r(m)ro  (cos  9  +  y~i  sin  9)m 


(a2  +  J2)1 


— i— ~  (cos  m0  +  y/-  1  sin  w0) , 


(a2  +  by 
Hence,  equating  real  and  imaginary  parts,  we  have 


e"uxcos  bxxm'xdx  =        v'"7  -  cos  m9 
(a2  +  J2)' 

<f°xsin  fo^™-1^  = — '—  sin  mO  \ 

Jo  (a3+ft2)T  J 


r(m) 

(66) 


in  which  9  =  tan 


-©■ 


7T 


If  we  make  a  =  o.  9  becomes  -,  and  these  formulae  become 

2 


*  For  a  rigorous  proof  of  the  validity  of  this  transformation  the  student  is 
referred  to  Serrett's  Gale.  Int.,  p.  194. 


Gamma  Functions.  167 

cos  o^«m_1  o$  =     7V    '  cos  — , 
Jo  bm  2      I 

n7    N  >•  (67) 

f      •     t     «,  !  7        r(m)    .    m7r    I 
sm  bxxm-1  dx  =  — £-^  sin  — 

Jo  5W  2       J 

It  may  be  observed  that  these  latter  integrals  can  be  ar- 
rived at  in  another  manner,  as  follows  : — 
From  (52)  we  have 

7        .00 
T  {n)  — —  =        e'zx  xn~x  cos  bz  dx ; 

z  Jo 

■ ,  ,  f  *  cos  &3  dz      I  "  f  "  ,         .  ,    , 

•'•  rW        » — =  e~zx  cos  bzz^dxdz. 

But,  by  (32),  we  have 

-co 

Jo  fr^o2' 


f  °°  cos  bzdz  _     1      f w    #n  db 
Jo    ~~ ^       ~I>)Jo    PT^2 


6* 


7T 


r(n)  rnr    1     /allv 

v  '  2  cos—  >  by  (27), 


in  which  w  must  be  positive  and  <  1 . 
In  like  manner  we  find 


sin  bzdz       b^1         tt 

0        zn      ~  r(n)       .    ^tt 

w  2  sm  — 


The  results  in  (67)  follow  from  these  by  aid  of  the  relation 
contained  in  equation  (55). 


168  Definite  Integrals. 

Examples. 

r„  +  ,)r("l±i) 

rigm-i(i  -  a;)"-1^  r(m)r(n) 

a'      Jo      (o  +  *)m+n     '  "    ««(i+«)»r(»w  +  «)' 

3.  Prove  that 

r1      x*dx         f1       dx  v 

J 0(1  -  z*)i     Jo(i  +  a*)*      2^2* 

.  L  r(n+i)coi 

4.  jo    C08(W)&.  „      - 

5-        I       /  -*  » 

J  0  y/  1  -  x» 

C"  embx  ,  tr 

b.  dx.  „     - 

Jo       x  "2 


(-:) 


"  '(H) 


123.  \umerieal  Calculation  of  Gamma  Func- 
tions.— The  following  Table  gives  the  values  of  log  r(p), 
to  six  decimal  places,  for  all  values  of  p  between  1  and  2 
(taken  to  three  decimal  places). 

It  may  be  observed  that  we  have  r(i)  =  r(2)  =  i,  and 
that  for  all  values  of  jp  between  1  and  2,  T(p)  is  positive  and 
less  than  unity ;  and  hence  the  values  of  log  T  (p)  are  negative 
for  all  such  values.  Consequently,  as  in  ordinary  trigono- 
metrical logarithmic  Tables,  the  Tabular  logarithm  is  obtained 
by  adding  10  to  the  natural  logarithm.  The  method  of 
calculating  these  Tables  is  too  complicated  for  insertion  in 
an  elementary  Treatise. 


Log 

•r(p) 

• 

p 

0 

1 

2 

3 

4 

5 

6 

7 

8 

•1 

I.OO 

975o 

9500 

9251 

9003 

8755 

8509 

8263 

8017 

7773 

r.oi 

9-997529 

7285 

7043 

6801 

6560 

6320 

6080 

5841 

5602 

5365 

i. 02 

5128 

4892 

4656 

4421 

4187 

3953 

372i 

3489 

3257 

3026 

1.03 

2796 

2567 

2338 

2110 

1883 

1656 

!43Q 

1205 

0981 

0775 

1.04 

0533 

0311 

0089 

9868 

9647 

9427 

9208 

8989 

8772 

8554 

1.05 

9.988338 

8122 

7907 

7692 

7478 

7265 

7052 

6841 

6629 

6419 

I  1.06 

6209 

6000 

5791 

5583 

5378 

5169 

4963 

4758 

4553 

4349  i 

1.07 

4145 

3943 

3741 

3539 

3338 

3138 

2939 

2740 

2541 

2344  l 

1.08 

2147 

i95i 

1755 

1560 

1365 

1172 

0978 

0786 

2594 

2403  1 

1.09 

0212 

0022 

9833 

9644 

9456 

9269 

9082 

8900 

8710 

8525 

1. 10 

9.978341 

8i57 

7974 

7791 

7610 

7428 

7248 

7068 

6888 

6709 

1. 11 

6531 

6354 

6177 

Logo 

5825 

5650 

5475 

53oi 

5128 

4955 

1. 12 

4783 

4612 

444 1 

4271 

4101 

3932 

3764 

3596 

3429 

3262 

113 

3096 

2931 

2766 

2602 

2438 
0835 

2275 

2113 

1951 

1790 

1629 

1.14 

1469 

1309 

"50 

0992 

0677 

0521 

0365 

0210 

0055 

1.15 

9.969901 

9747 

9594 

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9290 

9139 

8988 

8838 

8688 

8539 

1.16 

8390 

8243 

8096 

7949 

7803 

7658 

7513 

7369 

7225 

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1.17 

6939 

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6374 

6234 

6095 

5957 

5818 

5681. 

1. 18 

5544 

5408 

5272 

5137 

s™2- 

4868 

4734 

4601 

4469 

4337 

1.19 

4205 

4075 

3944 

3815 

3686 

3557 

3429 

3302 

3175 

3048 

i 
1.20 

2922 

2797 

2672 

2548 

2425 

2302 

2179 

2057 

1936 

1815 

1. 21 

1695 

1575 

1456 

1337 

1219 

IIOI 

0984 

0867 

0751 

0636 

1  1.22 

0521 

0407 

0293 

0180 

0067 

9955 

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9732 

9621 

95" 

I.23 

9.959401 

9292 

9184 
8128 

9076 

8968 

8861 

8755 

8649 

8544 

8439 

I.24 

8335 

8231 

8025 

7923 

7821 

7720 

7620 

752o 

7420 

1.25 

732i 

7223 

7125 

7027 

6930 

6834 

6738 

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6547 

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1.26 

6359 

6267 

6i73 

6081 

5989 

5898 

5807 

57i6 

5627 

5537 

1.27 

5449 

536o 

5273 

5185 

5099 

5OI3 

4927 

4842 

4757 

4673 

1.28 

4589 

45o6 

4423 

4341 

4259 

4178 

4097 

4017 

3938 

3858 

I.29 

3780 

3702 

3624 

3547 

3470 

3394 

33i8 

3243 

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3020 

2947 

2874 

2802 

2730 

2659 

2588 

2518 

2448 

2379 

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2310 

2242 

2174 

2106 

2040 

1973 

1007 

1842 

1777 

1712 

1.32 

1648 

1585 

1522 

1459 

1397 

1336 

1275 

1214 

"54 

1094 

1-33 

io35 

0977 

0918 

0861 

0803 

0747 

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0634 

0579 

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i-34 

0470 

0416 

0362 

0309 

0257 

0205 

^'53 

0102 

0051 

0001 

i-35 

9-949951 

9902 

9853 

9805 

9757 

9710 

9b63 

9617 

957i 

9525  | 

1.36 

9480 

9435 

939i 

9348 

9304 

9262 

9219 

9178 

9136 

9095 

i-37 

9054 

9015 

8975 

8936 

8898 

8859 

8822 

8785 

8748 

8711 

1.38 

8676 

8640 

8605 

857i 

8537 

8503 

8470 

8437 

8405 

8373 

i-39 

8342 

8311 

8280 

8250 

8221 

8192 

8163 

8i35 

8107 

8080 

1.40 

8053 

8026 

8000 

7975 

7950 

7925 

7901 

7877 

7854 

7831 

1.41 

7808 

7786 

7765 

7744 

7723 

7703 

7683 

7664 

7645 

7626 

1.42 

7608 

7590 

7573 

7556 

7540 

7524 

7509 

7494 

7479 

7465 

i-43 

745i 

7438 

7425 

7413 

7401 

7389 

7378 

7368 

7357 

7348 

1.44 

7338 

7329 

732i 

7312 

7305 

7298 

7291 

7284 

7278 

7273 

i-45 

7262 

7263 

7259 

7255 

7251 

7248 

7246 

7244 

7242 

7241 

1.46 

7240 

7239 

7239 

7240 

7240 

7242 

7243 

7245 

7248 

7251 

1.47 

7254 

7258 

7262 

7266 

7271 

7277 

7282 

7289 

7295 

7302 

1.48 

73io 

7317 

7326 

7334 

7343 

7353 

7363 

7373 

7384 

7395 

1.49 

7407 

7419 

743i 

7444 

7457 

747i 

7485 

7499 

75H 

7529 

Log 

?r(p). 

p 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.50 

9-947545 

756i 

7577 

7594 

7612 

7629 

7647 

7666 

7685 

7704 

«.$! 

7724 

7744 

7764 

7785 

7806 

7828 

7850 

7873 

7896 

7919 

1-52 

7943 

7967 

7991 

8016 

8041 

8067 

8093 

8120 

8146 

8174 

1.53 

8201 

8229 

8258 

8287 

8316 

8346 

8376 

8406 

8437 

8468 

1.54 

8500 

S532 

8564 

8597 

8630 

8664 

8698 

8732 

8767 

8802  J 

i-55 

8837 

8873 

8910 

8946 

8983 

9021 

9059 

9097 

9i35 

9i74  1 
9586  | 

1.56 

9214 

9254 

9294 

9334 

9375 

9417 

9458 

9500 

9543 

i-57 

9629 

9672 

9716 

8761 

9806 

9851 

9896 

9942 

9989 

0035  j 

1.58 

9.950082 

0130 

0177 

0225 

0274 

0323 

0372 

0422 

0472 

0522  1 

1.59 

0573 

0624 

0676 

0728 

0780 

0833 

0886 

0939 

o993 

1047 

1.60 

1 102 

if  57 

1212 

1268 

1324 

1380 

1437 

1494 

1552 

1610 

1.61 

1668 

1727 

1786 

1845 

1905 

1965 

2025 

2086 

2147 

2209 

1.62 

2271 

2333 

2396 

2459 

2522 

2586 

2650 

2715 

2780 

2845 

1.63 

2911 

2977 

3043 

3110 

3177 

3244 

33i2 

338o 

3449 

35i7 

1.64 

3587 

3656 

3726 

3797 

3867 

3938 

4010 

4081 

4154 

4226 

1.65 

4299 

4372 

4446 

4519 

4594 

4668 

4743 

4819 

4894 

4970 

1.66 

5047 

5124 

5201 

5278 

5356 

5434 

5513 

5592 

5671 

5740 

1.67 

5830 

59ii 

599i 

6072 

6154 

6235 

f>w 

6400 

6482 

6566 

1.68 

6649 

6733 

6817 

6901 

6986 

7072 

7157 

7243 

7322 

7416 

1.69 

7503 

7590 

7678 

7766 

7854 

7943 

8032 

8122 

8211 

8301 

1.70 

8391 

8482 

8573 

8664 

8756 

8848 

8941 

9034 

9127 

9220 

1.71 

9314 

9409 

9502 

9598 

9693 

9788 

9884 

9980 

0077 

0174 

1.72 

9.960271 

0369 

0467 

0565 

0664 

0763 

0862 

0961 

1061 

1 162 

i-73 

1262 

1363 

1464 

1566 

1668 

1770 

1873 

1976 

2079 

2183 

1.74 

2287 

2391 

2496 

2601 

2706 

2812 

2918 

3024 

3i3i 

3238 

i-75 

3345 

3453 

356i 

3669 

3778 

3887 

3996 

4105 

4215 

4326 

1.76 

4436 

4547 

4659 

4770 

4882 

4994 

5107 

5220 

5333 

5447 

1.77 

556i 

5675 

5789 

59o4 

6019 

6135 

6251 

6367 

6484 

6600 

1.78 

6718 

6835 

6953 

7071 

7189 

7308 

7427 

7547 

7666 

7787 

1.79 

7907 

8023 

8149 

8270 

8392 

8514 

8636 

8759 

8882 

9005 

1.80 

9129 

9253 

9377 

9501 

9626 

9751 

9877 

0008 

6129 

0255 

1.81 

9-970383 

0509 

0637 

0765 

0893 

1021 

1150 

1279 

1408 

1538 

1.82 

1668 

1798 

1929 

2060 

2191 

2322 

2454 

2586 

2719 

2852 

1.83 

2985 

3118 

3252 

3386 

3520 

3655 

3790 

3925 

4061 

4r97 

1.84 

4333 

447o 

4606 

4744 

4881 

5019 

5157 

5295 

5434 
6838 

5573 

1.85 

5712 

5852 

5992 

6132 

6273 

6414 

6555 

6697 

6980 

1.86 

7123 

7266 

7408 

7552 

7696 

7840 

7984 

8128 

8273 

Hi9 
9887 

1.87 

8564 

8710 

8856 

9002 

9149 

9296 

9443 

959i 

9739 

1.88 

9.980036 

9184 

0333 

0483 

0633 

0783 

0933 

1084 

1234 

1386 

1.89 

1537 

1689 

1841 

1994 

2147 

2299 

2453 

2607 

2761 

2915 

1.90 

3069 

3224 

3379 

3535 

3690 

3846 

4003 

4159 

43i6 

4474 

1.91 

4631 

4789 

4947 

5105 

5264 

5423 

5582 

5742 

5902 

6062 

1.92 

6223 

6383 

6544 

6706 

6867 

7029 

7192 

7354 

7517 

7680 

I  93 

7844 

8007 

8171 

8336 

8500 

8665 

8830 

8996 

9161 

9327 

1.94 

9494 

9660 

9827 

9995 

0162 

0330 

0498 

0666 

0835 

1004 

i-95 

9-99H73 

1343 

1512 

1683 

1853 

2024 

2195 

2366 

2537 

2709 

1.96 

2881 

3054 

3227 

3399 

3573 

3746 

3920 

4094 

4269 

4443 

1.97 

4618 

4794 

4969 

5145 

5321 

5498 

5674 

585f 

6029 

6206 

1.98 

6384 

6562 

6740 

6919 

7098 

7277 

7457 

7637 

7817  7997 

1.99 

8178  8359 

8540 

8722 

8903 

9085 

9268 

945o 

9633  |  9816  | 

Examples.  171 

Examples. 

f«     dx  ._ 

r-       I       / .  Ans.  2y«, 

2.  lif{x)  =/(»  +  x)  for  all  values  of  x,  prove  that 

rna  ca 

\    f{x)dx  =  n  l   f(x)dx, 
Jo*  Jo 

where  n  is  an  integer. 


Jo 


y  m  -  #* 

r2         dx 

4-       I    ■ 

J\X  \Zx2  —  I 

5.       I    sin-1^^. 
Jo 


f1  dx «■ 

J°(i  +*)a/i  +  2ic-«2  '     4  a/2 

f  °°             dx  ir 

I       ; :,    ao  -  i2  heing  positive.  ,,         . 

J  ..  a  +  zbx  +  cz»                       e  *  y^  _  $2 


8.  Prove  that 

dx 


Jdx  v  . 

r— 

Jo   1  + 


9 


Jo  I  + 


cos  0  COS  # 
dx 


COS  0  COS  » 


IT 

12  dx 
0  a2  sin2  a;  +  £2cos2#* 


f»  dx 

I2#       Jo  (a2 sin2 x  +  b* cos2 xf 


Ans. 

sin0* 

>» 

0 
sine" 

7T 

j> 

2ab' 

7T(«2  +  *2) 

»» 

4«3  A3 

172  Definite  Integrals. 


TT2fl2         a* 


'3-     j    i/a*  -  x1  ™ 


.4.  r- 

J-i  (a 


a  16         4 


-,  a  >  *. 


(a-&r)<\/i-s2  </a2-£2 


.5.  r 

J  *  y/  (x  -  a)  Q8  -  a:) 


16. 


r«+Va»-6»  (y2  +  £2)yrfy 


01-  XT.    i.     f        Sm  "*   C0S  OX    .  TT  ,.  , 

7.     Snow  that  1     dx  =  -,  or  o,  according  as  a  >  or  <  b ;  and 


that  when  a  =  b  the  value  of  the  integral  is  -. 

4 


f+1  *■  1     1      fi  +  y/ab\ 

18.  ,  ,  «£<  1.  .4n*.  ■   ,_log    — _  • 

J-iv/Ci-zas  +  ^Xl-^te  +  a2)'  ^J        \i-*/ab) 

IT 

19.  1*  tan6£<fc.  „    -flog2 — J. 


J  4     sinxw 
— .  „     -  +  tan-1 
0    1  +  cos-2  x  4 


y= 


2 1 .  If  every  infinitesimal  element  of  the  side  e  of  any  triangle  he  divided 
by  its  distance  from  the  opposite  angle  Ct  and  the  sum  taken,  show  that  its 
value  is 


log  [  cot  —  cot  —  J  . 


22.  Being  given  the  base  of  a  triangle  ;  if  the  sum  of  every  element  of  the 
base  multiplied  by  the  square  of  the  distance  from  the  vertex  be  constant,  show 
that  the  locus  of  the  vertex  is  a  circle. 


J  a  cos4 
.77 


i      tan-'* 
A.ns.  — 


e2  cos2  0  e2        e3 


f  2    cos20  sin  9^0  \/ 1  +  e2      log  (e  +  */ \  +  e1 

24'     Jo  v/i  +  ^cosV  "  2*3  2^ 


Examples.  173 

25.  Deduce  the  expansions  for  sin  a;  and  cos  x  from  Bernoulli's  series. 

26.  Show  that  the  integral 

r1 

>dx 


£w(log#)' 
J  0 


can  be  immediately  evaluated  by  the  method  of  Art.  in,  when  m  is  an  integer. 

f00  tmrl{ax)dx  .      -k .      . 

27-           — ,         ox  -Arts.  -  log  (1  +  a). 

Jo     x(i  +  x2)  2 


•(1  +  *2) 
28.  Find  the  value  of 


log  (1  -  2a  cos  x  +  «2)  fifc, 


distinguishing  between  the  cases  where  a  is  >  or  <  1. 

Am.  a  <  1,  its  value  is  o. 
,,     a  >  I,  its  value  is  2tt  log  a. 

29.  If  /  (#)  can  be  expanded  in  a  series  of  the  form 

ao  +  a\  cos  x  +  a%  cos  2X  +  .  .  .  +  <rn  cos  w#  +  .  .  .  , 

show  that  any  coefficient  after  a0  can  be  exhibited  in  the  form  of  a  definite 
integral. 

Am.  an  —  —  I  /(«)  cos  nxdx. 

IT  JQ 

30.  Find  the  analogous  theorem  when  f(x)  can  be  expanded  in  a  series  of 
of  multiples  of  x ;  and  apply  the  method  to  prove  the  relation 


(sin  2X      sin  $x      „     \ 
sin  # + &c.  J 1 
2                    3/' 


when  x  lies  between  +  ir. 

31.  Prove  the  relation 

IT  JT 

32.  Express  the  definite  integral 

fa         <# 

Jo, 


1  \/ 1  —  /c2sin2 
in  the  form  of  a  series,  /c  being  <  1. 
2 


^i('*(;)*+WH^)W 


174  Definite  Integrals. 

fFlog(i+cos«cos*)tf*  ^.I/l?_A 

**      Jo  cos*  2\4  / 

34.      I     xtr^coabxdx,  where  a  >  o. 

35-  J.  — s — ix- 

IT 

36.      f  2  log  (a2  cos2  6  +  j82  sin2  0)  dd. 

it 

cl,      /a  +  bBmd\    dd             ,  .    1/b\ 

37-         log[ r^-^; )  3TTi  a  >  ••  »    TSinM-J. 

Jo    6  V«-*8me/  *me  W 

f1        tf* 
3  '     Jo  (t  _  a*)*' 


>> 

(a2  +  ty 

>> 

2      S  (      a«0P 

>» 

,      o  +  /3 
,log__. 

IT 


f1         <£r 
39-  1- 

Jo(i-^r 


jr 


ff        cosr#<fo 

40.  

Jo  1  —  20  COS  X  + 


••         coarxdx  irar 

*'  "     i-a* 


41.  Find  the  sum  of  the  series 


n  n  n  n 

+    .      .»  +    o  .     o  +  •  •  •  +  — ;• 


n2  +  i2      n*  +  22      »2  +  32  2n2 

when  n  is  increased  indefinitely. 

This  is  evidently  represented  hy  the  definite  integral 

r1      dx  IT 

-,  or  =  -. 

J  0  1  +  *s  4 

42.  Find  the  limit  of  the  sum 

111  1 

v/»2  -  I2      \/»2  "  2'      V  «2  -  32  \A2  -  (n  -  1)* 


when  w  =  00.  -4«*.  — . 

2 


Examples.  175 

43.  Prove  that 

fi~  ,        »»(m  —  1)  f2  ,  , 

coswo;  cos  nx  dx  =  -4 tt       cosm_2  x  cos  nxdx  ; 

Jo  tw58  -  w2    J o 

and  hence,  deduce  the  values  of  the  integrals 

IT  7T 

cos2m#  cos  (2n  +  i)  x  dx,  and  1    cos-m+1  x  cos  mx  dx, 
when  m  and  «  are  integers. 


7T«» 


44.  I      log(i  -  2a  cos  0  +  a2)  cos  nOdd,  when  a2  <  1.        Ans 

45.  cos  — tf#.  'Js^s~^->       c  ,,     1. 

J  -co  2 

f1log(l  +  *)  ir 

47.  Prove  the  following  equation  : 

f«  ^0  I  f7T   (  .        , 

I     ; - sr-  ■  . — -5        (1  -  24  COS0  +  a2)n-1^0. 

Jo  (i-2«cos0  +  «2)«      (i-«-)"-1Jo  * 

48.  Prove  the  more  general  equation 

It  siamdde  1  fw  sinm0^0 

0  (1  -  za  cos  0  +  a2)n  ~  (1  -  a2)2" "»»-1  ]0  (1  -  2a  cos0  +  «2)1+'«-*' 

in  which  m  +  1  is  positive. 


(    176    ) 


CHAPTER  VII. 


AREAS   OF   PLANE   CURVES. 


126.  Areas  of  Curves. — The  simplest  method  of  regarding 
the  area  of  a  curve  is  to  suppose  it  referred  to  rectangular 
axes  of  co-ordinates;  then,  the  area  included  between  the 
curve,  the  axis  of  #,  and  the  two  ordinates  corresponding  to  the 
values  x0  and  xx  of  #,  is  represented  by  the  definite  integral 


ydx. 


For,  let  the  area  in  question  be  represented  by  the  space 
AB  FT,  and  suppose  B  V  divided  into  n  equal  intervals,  and 
the  corresponding  ordinates  drawn,  ( 
as  in  the  accompanying  figure. 

Then  the  area  of  the  portion 
PMNQ  is  less  than  the  rectangle 
pMNQ,  and  greater  than  PMNq. 

Hence  the  entire  area  AB  FT  is 
less  than  the  sum  of  the  rectangles 
represented  hypJUNQ,  and  greater 
than  the  sum  of  the  rectangles 
PMNq  ;  but  the  difference  be- 
tween these  latter  sums  is  the  sum 
of  the  rectangles  Pp  Qq,  or  (since  the  rectangles  have  equal 
bases)  the  rectangle  under  MN  and  the  difference  between 
TV  and  AB.  Now,  by  supposing  the  number  n  increased 
indefinitely,  MN  can  be  made  indefinitely  small,  and  hence 
the  rectangle  MN  {TV -  AB)  also  becomes  infinitely  small. 
Consequently  the  difference  between  the  area  ABVT  and 
the  sum  of  the  rectangles  PMNq  becomes  evanescent  at  the 
same  time. 


Areas  of  Curves.  177 

If  now  the  co-ordinates  of  P  be  denoted  by  x  and  y,  and  MN 
by  Aa?,  it  follows  that  the  area  AB  VT  is  the  limiting  value* 
of  2(y  Ax)  when  the  increment  Ax  becomes  infinitely  small  ; 

or  area  AB  VT  =       y  dx ;  where  xx  =  0  J7,  #0  =  0J?. 

It  should  be  observed  that  this  result  requires  that  y 
continue  finite,  and  of  the  same  sign,  between  the  limits 
of  integration. 

If  y  change  its  sign  between  the  limits,  i.e.  if  the  curve 
cut  the  axis  of  x,  the  preceding  definite  integral  represents 
the  difference  of  the  areas  at  opposite  sides  of  the  axis  of  x. 

In  such  cases  it  is  preferable  to  consider  each  area  sepa- 
rately, by  dividing  the  integral  into  two  parts,  separated  by 
the  value  of  x  for  which  y  vanishes. 

The  preceding  mode  of  proof  obviously  applies  also  to 
the  case  where  the  co-ordinate  axes  are  oblique ;  in  which 
case  the  area  is  represented  by 

sin  iu       y  dx, 


J  *0 


where  w  represents  the  angle  between  the  axes. 

In  applying  these  formulae  the  value  of  y  is  found  in 
terms  of  x  by  means  of  the  equation  of  the  curve  :  thus, 
if  y  =/(x)  be  this  equation,  the  area  is  represented  by 

^f(x)dx, 

taken  between  suitable  limits. 

Conversely,  the  value  of  any  definite  integral,  such  as 

}af(x)dx, 

may  be  represented  geometrically  by  the  area  of  a  definite 
portion  of  the  curve  represented  by  the  equation 

V  =/0*)- 

*  This  demonstration  is  substantially  that  given  by  Newton  (see  Principia, 
Lib.  I.,  Sect.  1.,  Lemma  2) ;  and  is  the  geometrical  representation  of  the  result 
established  in  Art.  90. 

The  modification  in  the  proof  when  the  elements  of  BV  are  considered 
unequal,  but  each  infinitely  small,  is  easily  seen.  It  may  be  remarked  that  the 
result  here  given  is  but  a  particular  case  of  the  general  principle  laid  down  in 
Arts.  38,  39,  ■&&■  Calc. 

[12] 


178 


Areas  of  Plane  Curves. 


On  account  of  this  property  the  process  of  integration  was 
called,  by  Newton  and  the  early  writers  on  the  Calculus, 
the  method  of  quadratures. 

Again,  it  is  plain  that  the  area  between  the  curve,  the 
axis  of  y,  and  two  ordinates  to  that  axis,  is  represented  by 

jxdy, 

taken  between  the  proper  limits :  the  co-ordinate  axes  being 
supposed  rectangular. 

We  proceed  to  illustrate  this  method  of  determining 
areas  by  a  few  applications,  commencing  with  the  simplest 
examples. 

127.  The  Circle. — Taking  the  equation  of  a  circle  in 
the  form 


x2  +  y2  =  a2,  we  get  y  =  */a2  -  x\ 
and  the  area  is  represented  by 


J  v  a1  -  x2dx, 


taken  between  proper  limits. 

For  instance,  to  find  the  area  of 
the  portion  represented  by  APDJE 
in  the  accompanying  figure.  Let 
x  =  a  cos  0,  then  the  area  in  ques- 


Fig.  2. 


tion  plainly  is  represented  by 

fa  a1 

a2       sin2  OdQ  -  —  (a  -  sin  a  cos  a) ;  where  a  =  L  DC  A. 

This  result  is  also  evident  from  geometry ;  for  the  area 
DPAE  is  the  difference  between  DP  AC  and  DCE,  or  is 


The  area  of  the  quadrant  ACB  is  got  by  making  a 


7rflr 


and  accordingly  is  — :  hence  the  entire  area  of  the  circle 


The  Ellipse.  179 

128.  The  Ellipse. — From  the  equation  of  the  ellipse 

x*     y1  ,         b     ' 

-+|2=i,  wegety  =  -^  _**, 

and  the  element  of  area  is 

b      . 

~  va?  -  x2dx\ 

b 
but  this  is  -  times  the  area  of  the  corresponding  element  of 

a 
the  circle  whose  radius  is  a :  consequently  the  area  of  any 

portion  of  the  ellipse  is  -  times  that  of  the  corresponding  part 

a 

of  the  circle.     This  is  also  evident  from  geometry. 
The  area  of  the  entire  ellipse  is  irab. 
Again,  if  the  equation  of  an  ellipse  be  given  in  the  form 

Ax%  +  By1  =  C,  its  area  is  evidently 

V  AB 

As  an  application  of  oblique  axes,  let  it  be  proposed 
to  find  the  area  of  the  segment 
of  an  ellipse  cut  off  by  any  chord 
BIT. 

Draw  the  diameter  AA\  con- 
jugate to  the  chord,  and  BB' 
parallel  to  it.  Then,  C  being 
the  centre,  let 


CA'  =  a\  CB'  =  V,ACB'  =  w, 


X"       »* 


and  the  equation  of  the  ellipse  is  —%  +  ^  =  1 ;  henoe  the  area 

BA!If  is  represented  by 

b'    .         CCA'     j 

2  -  sin  u)  \       va'2  -  x2dx  =  db'  sin  w  (a  -  sin  a  cos  a), 
a  j  ce 

CE 
where  cos  a  =  -^-r,. 

Again,  a'  V  sin  w  =  ab,  by  an  elementary  property  of  the 
ellipse,  a  and  b  being  the  semiaxes. 

Hence  the  area  of  the  segment  in  question  is 

ab(a  -  sin  a  cos  a). 
[12  a] 


180 


Areas  of  Plane  Curves. 


This  result  can  also  be  deduced  immediately  from  the 
circle  by  the  method  of  orthogonal  projection. 

It  may  be  observed  that  if  we  denote  the  area  of  an  elliptic 
sector,  measured  from  the  axis  major  to  a  point  whose  co- 
ordinates are  x,  y,  by  S,  we  may  write 


x  iB 

-  =  COS  — -  =  COS  a, 

a  ab 


r  =  sin  — r  =  sin  a. 
o  ab 


129.  The  Parabola. — Taking  the 
equation  of  the  parabola  in  the  form 

y2  =  px,  we  get  y  =  \/px. 

Hence  the  area  of  the  portion  APN  is 

f  2,2 

pk     x^dx,  or  -  p$x*,  i.e.  -  xy. 

J  o  o 

Consequently,  the  area  of  the  seg- 
ment PAP',  cut  off  by  a  chord  perpen- 
dicular to  the  axis,  is  f  of  the  rectangle 
PMM'F.  S4' 

It  is  easily  seen  that  a  similar  relation  holds  for  the  seg- 
ment cut  off  by  any  chord. 

More  generally,  let  the  equation  of  the  curve  be  y  =  ax", 
where  n  is  positive. 

f  f  axnn 

Here  ydx  =  a    x"dx  = +  const. 

J*  J  n+i 

If  the  area  be  counted  from  the  origin,  the  constant 

vanishes,  and  the  expression  for  the  area  becomes 


or 


xy 


n  +  1  n  +  1 

Hence,  the  area  is  in  a  constant  ratio  to  the  rectangle 
under  the  co-ordinates.  A  corresponding  result  holds  for 
oblique  axes.  The  discussion,  when  n  is  negative,  is  left  to 
the  student. 

Example. 

Express  the  area  of  a  segment  of  a  parabola  cut  off  by  any  focal  chord  in 
terms  of  /,  the  length  of  the  chord,  and  p,  the  parameter  of  the  parabola. 

$pi 
Am.    -r. 


The  Hyperbola. 


181 


130.  The  Hyperbola. — The  simplest  form  of  the 
equation  of  a  hyperbola  is  where  the  asymptotes  are  taken 
for  co-ordinate  axes ;  in  this  case  its  equation  is  of  the  form 


xy  =  c2. 


Hence,  denoting  the  angle  between  the  asymptotes  by  w, 
the  area  between  the  curve  and  an  asymptote  is  denoted  by 

'  dx 


&  sin  (o\  — ,  or  cr  sin  w  log  I  — 

J   &  \Xq 


where  xx  and  x0  are  the  abscissae  of  the  limiting  points. 
If  the  curve  be  referred  to  its  axes,  its  equation  is 


a2     b2 


and  the  element  of  area  ydx  becomes 

-  vx2  -  a2dx. 
a 

Hence  the  area  is  represented  by 

-    */x2  -  a2dx, 
taken  between  proper  limits. 


Again,      */x2  -  a2dx  = 


W«? 


Also,  integrating  by  parts,  we  have 

J  yV  -  a2dx  =  x*/x2-a2- 
Adding,  and  dividing  by  2,  we  get 


\/x2  -  a2 


x2dx 


's/x2  -  a2 


2 
x*/x2 


? 


dx 


V: 


x~-af 


log  {x  +  */x2-a2 


182  Areas  of  Plane  Curves. 

Accordingly,  if  we  suppose  the  area  counted  from  the 
summit  A,  we  have 

Again,  since  the  triangle  CPN  =  \xy,  it  follows  that 
sector  ACP^  log  (j  +  |\ 

For  a  geometrical  method  of  finding  the  area  of  a  hyper- 
bolic sector,  see  Salmon's  Conies,  Art.  395. 

130(a).  Hyperbolic  Sine  and  Cosine. — If  8  repre- 
sent the  sector  ACP,  the  final  equation  of  the  preceding 
Article  becomes 

which  may  also  be  written 

•      V 
a      0 
introducing  a  single  letter  v  to  denote  the  quantity 

25 


*m* 


ab 


Hence,  by  the  equation  of  the  hyperbola,  we  get 

-  -  I  -  g* 

a      0 

Thus,  in  analogy  with  the  last  result  of  Art.  128,  calling  the 
following  functions  the  hyperbolic  cosine  and  hyperbolic 
sine  of  v,  and  for  brevity  writing  them  cosh  v,  and  sinh  v, 

ev  +  e~v  =  2  cosh#,     ev  -  e"°  =  2  sinhp,  (2) 

the  co-ordinates  of  any  point  on  the  curve  are 

x  .  ,  28     y       .  .  .  .  28 

-  =  cosh  v  =  cosh  — r,    -  =  smh  v  =  Sinn  -r-. 
a  ab      b  ab 


The  Catenary. 


183 


"We  might  have  treated  the  matter  differently  by  intro- 
ducing the  angle  <p  denned  by  the  equation  x  =  a  sec  0,  and 
therefore  y  =  b  tan  0  (for  the  geometric  meaning  of  this 
transformation,  see  Salmon's  Conies,  Art.  232);  whence  (1) 
may  be  written* 

|g-,-logfang  +  f): 

and  we  see  that  the  hyperbolic  cosine  of  a  real  quantity  is  the 
secant,  and  the  hyperbolic  sine  the  tangent  of  the  same  real 
angle.     Also,  since 

sinh  v  i,i  cosh  v 

Sin  6  = : ,  COS  6  =  : ,  COt  <b  =   -r-i ,   C0S6C  6  =  -7—, , 

cosh  v  cosh  v  sinn  v  T     sinh  v 

we  can  obviously  extend  the  names  of  the  other  trigonometrical 
functions  likewise.  Again,  putting  in  (2)  for  0,  u\/  -  1,  or 
iuy  they  become,  by  Art.  8, 

cos  u  =  cosh  iu,     i  sin  u  =  sinh  iu. 

131.  The  Catenary. — If  an  inelastic  string  of  uniform 
density  be  allowed  to  hang  freely  from  two  fixed  points,  the 
curve  which  it  assumes  is  called  the  Catenary. 

Its  equation  can  be  easily  arrived 
at  from  elementary  mechanics,  as  fol- 
lows : — 

Let  V  be  the  lowest  point  on  the 
curve;  then  any  portion  VP  of  the 
string  must  be  in  equilibrium  under 
the  action  of  the  tensions  at  its  ex- 
tremities, and  its  own  weight,  W.  F-    ^ 

Let  A  be  the  tension  at  V;  T  that 
at  P,  which  acts  along  PR,  the  tangent  at  P;  lPRM  =  0. 
Then,  by  the  property  of  the  triangle  of  force,  we  have 

W:A  =  PM:RM\ 
.*.   W  =  A  tan0. 


*  When  <p  is  related  to  v  by  this  equation,  <p  is  what  Professor  Cayley 
{Elliptic  Functions,  p.  56)  calls  the  gudermannian  of  v,  after  Professor  Guder- 
mann,  and  writes  the  inverse  equation  <p  =  gdv. 


.184 


Areas  of  Plane  Curves. 


Again,  if  s  be  the  length  of  VP,  and  a  that  of  the  portion 
of  the  string  whose  weight  is  A,  we  have,  since  the  string  is 
uniform, 

W=AS-x 


.',  s  =  a  tan  0. 

This  is  the  intrinsic  equation  of  the   catenary. 
Calc,  Art.  242  (a).) 

Its  equation  in  Cartesian   co- 
ordinates can  be  easily  arrived  at.     A\ 

For,  on  the  vertical  through  V      \ 
take  VO  =  rt,  and  draw  OX  in  the 
horizontal  direction,  and   assume 
OX  and  OF  as  axes  of  co-ordi- 
nates.    Let 


(Diff. 


then 


PN=y,     ON=x, 


Hence 


-r  =  tan  0, 
dx            r 

Fig. 

dy 

dx 

—  =  cos  6  ; 

ds           T 

dy      dy  ds 
d<j>      ds  d<p 

sin  d>      dx 

a  — .   9     ~r 
cos2  0      d(f> 

a 

COS0* 

y  =  a  sec  0, 

x  =  a  log  (sec 

0  +  tan 

*)•     (3) 


No  constant  is  added  to  either  integral,  since  y  =  a,  and 
x  =  o,  when  0  =  o. 

From  the  latter  equation  we  get 

sec  0  +  tan  0  =  ea ; 


also  sec  0  -  tan  0 

Hence,  we  have 

2  sec0 


sec  0  +  tan  0 

X  _X  X 

"  +  e~a,  2  tan  0  =  e~a 


=  e 


Examples.  185 

Consequently, 

a  (  x       -x\ 

y  =  -  [ea  + e  a}  (4) 

Also  s  =  -(ea-e  aj.  (5) 

In  the  notation  of  last  Article  these  equations  may  be 
written 

-  =  cosh  -  and  -  =  sinh  -. 
a  a         a  a 

Again,  if  NL  be  drawn  perpendicular  to  the  tangent  at 
P,  we  have 

NL  =  PN  cos  0  ;  .*.  NL  =  a.  (6) 

Also  PL  =  NL  tan  <j>;  ,\  PL  =  s  =  PV.  (7) 

The  area  of  any  portion  VPNO  is 


-  [*  (J  +  e"^  <&  =  -  (ea  -  /•]  -  a  (f  -  a*)K 


(8) 


Accordingly,  the  area  VPNO  is  double  that  of  the  triangle 
PNL. 

Examples. 

1.  To  find  the  area  of  the  oval  of  the  paranoia  of  the  third  degree  with  a 
double  point 

cy2  =  (x-a)(x-  b)2. 

.  A 

The  area  in  question  is  represented  by    q 

_^f6  / . 

/-I     (b  -  x)  v  x  —  adz.  _. 

Vc) aK  )W  Fig.  8. 

Let  z  —  a  =  zi.  and  we  easily  find  the  area*  to  be  — '-. 

2.  Find  the  whole  area  of  the  curve   a2y2  =  x3  {la  —  x).  Ans.  wa2. 

3.  Find  the  whole  area  between  the  cissoid  xz  =  y2  {a  -  x)  and  its  asymptote. 

2</  2 

*  The  student  will  find  little  difficulty  in  proving  that  this  area  is  — - — 

times  the  rectangle  which  circumscribes  the  oval,  having  its  sides  parallel  to  the 
co-ordinate  axes. 


186 


Since  x  -  a  =  o  is  the  equation 
presented  by 


Areas  of  Plane  Curves. 

the  asymptote  the  area  in  question  is  re- 
x^dx 


I  o  («  -  x)i 


Let  x  =  a  sin2  0,  and  this  becomes 


I.- 


2a2       sin*0tf0 


hence  the  area  in  question  is  |  ira2. 

o 

4.  Find  the  area  of  the  loop  of  the  curve 

azy2  =  x*(b  +  x). 

This  curve  has  been  considered  in  Art.  262,  Diff. 
Calc.  Its  form  is  exhibited  in  the  annexed  figure ;  and 
the  area  of  the  loop  is  plainly 


x2\/b  +  xdx. 


Let  b  +  x  =  z2,  and  it  is  easily  seen  that  the  area 
in  question  is  represented  by 

8.bi 


Fig.  9. 


3  •  5  •  7  •  <*a 

5.  Find  the  area  between  the  witch  of  Agnesi 
xy1  =  4a2  (2a  -  x) 
and  its  asymptote.  Ans.  4*0?. 

132.  In  finding  the  whole  area  of  a  closed  curve,  such  as 
that  represented  in  the  figure,  we 
suppose  lines,  PM,  QN,  &c,  drawn 
parallel  to  the  axis  of  y ;  then,  as- 
suming each  of  these  lines  to  meet 
the  curve  in  but  two  points,  and 
making  PM  =  y2f  P'M  =  yiy  the 
elementary  area  PQQ'P'  is  repre- 
sented by  (y2  -  yx)  dx,  and  the  en- 
tire* area  by 

COB' 

(y2-yi)dx;  Fig.  10. 

J  OB 

in  which  OB,  01?  are  the  limiting  values  of  x. 

*  This  form  still  holds  when  the  axis  of  x  intersects  the  curve,  for  the  ordi- 
nates  below  that  axis  have  a  negative  sign,  and  (1/2  -  y\)  dx  will  still  represent 
the  element  of  the  area  between  two  parallel  ordinates. 


MN 


1TX 


The  Ellipse.  187 

For  example,  let  it  be  proposed  to  find  the  whole  area  of 
an  ellipse  given  by  the  general  equation 

ax2  +  ihxy  +  by2  +  2gx  +  2fy  +  c  =  o. 
Here,  solving  for  y,  we  easily  find 


y*-yi  =  r  */{h2  -  ab)  x2  +  2  (hf-  bg)  x  +/2  -  be. 

Also,  the  limiting  values  of  x  are  the  roots  of  the  quadratic 
expression  under  the  radical  sign. 

Accordingly,  denoting  these  roots  by  a  and  /3,  and  observ- 
ing that  h2  -  ab  is  negative  for  an  ellipse,  the  entire  area  is 
represented  by 


2  v  ab  -  h 


-J  v/V-oX/3 


x)dx. 


b 

To  find  this,  assume    x  -  a  =  (j3  -  a)  sin2  6  ; 
then  fi-x  =  (j3-a)cos20, 

and  we  get 

IT 

1     */{x-a){$-x)  dx  =  2  (j3  -  a)2   [ '  Sin2 0  COS2  0  </0 

-f(0-«>-. 


Again,    (/3-a)2  =  4- 


{hf-bg)2 +{,/*- be)  {ab  -  h2 

{ab  -  h2y 

_  4b (a f2  +  bg2  +  eh2  -  ifgh  -  abc) 
Jab^h2)2 
Hence  the  area  of  the  ellipse  is  represented  by 
ir  {of2  +  bg2  +  ch2  -  2fgh  -  abc) 
{a~b  -  h2)$  * 

This  result  can  be  verified  without  difficulty,  by  deter- 
mining the  value  of  the  rectangle  under  the  semiaxes  of  an 
ellipse,  in  terms  of  the  coefficients  of  its  general  equation. 

It  is  worthy  of  observation  that  if  we  suppose  a  closed 
curve  to  be  described  by  the  motion  of  a  point  round  its  en- 
tire perimeter,  the  whole  inclosed  area  is  represented  by  j  ydx, 
taken  for  every  point  around  the  entire  curve. 


188  Areas  of  Plane  Curves. 

Thus,  in  the  preceding  figure,  if  we  proceed  from  A  to  A' 
along  the  upper  portion  of  the  curve,  the  corresponding  part 
of  the  integral  \ydx  represents  the  area  APA'&B.  Again, 
in  returning  from  A  to  A  along  the  lower  part  of  the  curve, 
the  increment  dx  is  negative,  and  the  corresponding  part 
of  I  ydx  is  also  negative  (assuming  that  the  curve  does  not 
intersect  the  axis  of  0),  and  represents  the  area  AFABI?, 
taken  with  a  negative  sign.  Consequently,  the  whole  area  of 
the  closed  curve  is  represented  by  the  integral  j  ydx,  taken 
for  all  points  on  the  curve. 

The  student  will  find  no  difficulty  in  showing  that  this 
proof  is  general,  whatever  be  the  form  of  the  curve,  and 
whatever  the  number  of  points  in  which  it  is  met  by  the 
parallel  ordinates. 

To  avoid  ambiguity,  the  preceding  result  may  be  stated  as 
follows  : — The  area  of  any  closed  curve  is  represented  by 

\yids 

taken  through  the  entire  perimeter  of  the  curve,  the  element  of  the 
vurve  being  regarded  as  positive  throughout. 

The  preceding  is  on  the  hypothesis  that  the  curve  has  no 
double  point.     If  the  curve  cut  itself,  so  as  to  form  two  loops, 

fdx 
y  —  ds,  when  taken  round  the  entire 

perimeter,  represents  the  difference  between  the  areas  of  the 
two  loops.  The  corresponding  result  in  the  case  of  three  or 
more  loops  can  be  readily  determined. 

133.  In  many  cases,  instead  of  determining  y  in  terms  of 
xt  we  can  express  them  both  in  terms  of  a  single  variable, 
and  thus  determine  the  area  by  expressing  its  element  in 
terms  of  that  variable. 

For  instance,  in  the  ellipse,  if  we  make  x  =  a  sin  0,  we 
get  y  =  b  cos  0,  and  ydx  becomes  db  cos2^  d<j>,  the  integral  of 
which  gives  the  same  result  as  before. 

In  like  manner,  to  find  the  area  of  the  curve 


©Ml)'- 


Let  x  =  a  sin30,  then  y  =  b  cos3<£,  and  ydx  becomes 
3a b  sin2  <j>  cos4  (fxlcp  : 


The  Cycloid. 
hence  the  entire  area  of  the  curve  is  represented  by 


189 


l: 


nab  J    sin20  cos40cfy>  =  -irab. 
Examples. 


r.  Find  the  whole  area  of  the  evolute  of  the  ellipse 


x6      v' 

-S  +  TS  -  I. 


Am. 


3*-Q2  -  V'-Y 
Sab 


2.  Find  the  whole  area  of  the  curve 
2 


(9T+  9"*- 


A,   ■■3.S-0»+').'.3-5...(»l'),A 
2.4.6 2(m  +  «+i) 

134.  The    Cycloid. — In   the   cycloid,   we   have    (Diff. 
Calc,  Art.  272), 

x  =  a  (0  -  sin  0),    y  =  a  (1  -  cos  9) ; 


ydx  =  a2 


(1  -  cos  0)2d0  =  4«2  [sin4-tf0. 


Taking  6  between  o  and  7r,  we  get  ^na2  for  the  entire 
area  between  the  cycloid  and  its  base. 

The  area  of  the  cycloid  admits  also  of  an  elementary 
geometrical  deduction,  as  follows : — 


^^     /          M 

^pr 

yS                           1                N 

/               (        c 

/                                  \           N 

T 

\? 

I 

V 

/                                        \        M 

yp' 

\ 

a:  D  A 

Fig.  11. 

It  is  obviously  sufficient  to  find  the  area  between  the 
semicircle  BPD  and  the  semi- cycloid  BpA.  To  determine 
this,  let  points  P  and  2*  be  taken  on  the  semicircle  such  that 
arc  BP  =  arc  DP* :  draw  MPp  and  M'P'p'  perpendicular  to 
BD.  Take  MN  and  M' N'  of  equal  length,  and  draw  Hq 
and  iVV,  also  perpendicular  to  BD :  then,  by  the  fundamen- 
tal property  of  the  cycloid,  the  line  Pp  =  arc  BP,  and  P'p' 
-  arc  BP' :     .*.  Pp  +  Pp'  =  semicircle  =  7r«. 


190 


Areas  of  Plane  Curves. 


Now,  if  the  interval  MN  be  regarded  as  indefinitely  small, 
the  sum  of  the  elementary  areas  PpqQ  and  P'p'q'Q  is  equal 
to  the  rectangle  under  MN  and  the  sum  of  Pp  and  P'p',  or  to 
ira  x  MN. 

Again,  if  the  entire  figure  be  supposed  divided  in  like 
manner,  it  is  obvious  that  the  whole  area  between  the  semi- 
circle and  the  cycloid  is  equal  to  ira  multiplied  by  the  sum  of 
the  elements  MN,  taken  from  B  to  the  centre  C,  i.e.  equal  to  ira2. 

Consequently  the  whole  area  of  the  cycloid  is  3na2,  as 
before. 

The  area  of  a  prolate  or  curtate  cycloid  can  be  obtained 
in  like  manner. 

135.  Areas  in  Polar  Co-ordinates. — Suppose  the 
curve  APB  to  be  referred  to  polar  co-ordinates,  0  being  the 
pole,  and  let  OP,  OQ,  OP  represent  consecutive  radii  vectores, 
and  PL,  QM,  arcs  of  circles  described  with  0  as  centre.  Then 
the  area  OPQ  =  OPL  +  PLQ ;  but 
PLQ  becomes  evanescent  in  com- 
parison with  OPL  when  P  and  Q 
are  infinitely  near  points;  conse- 
quently, in  the  limit  the  elemen- 

r2d0 
tary  area  OPQ=  area  OPL  = ; 

r  and  6  being  the  polar  co-ordi- 
nates of  P. 

Hence  the  sectorial  area  A  OB 
is  represented  by 


Fig.  12. 


dO, 


where  a  and  |3  are  the  values  of  0  corresponding  to  the  limit- 
ing points  A  and  B. 

136.  Area  of  Pedals  of  Ellipse  and  Hyperbola. — 

For  example,  let  it  be  proposed  to  find  the  area  of  the  locus 

of  the  foot  of  the  perpendicular  from  the  centre  on  a  tangent 

to  an  ellipse. 

&    y1 
Writing  the  equation  of  the  ellipse  in  the  form-  +  j-=  1, 

the  equation  of  the  locus  in  question  is  obviously 
r2  =  fl2cos30  +  b2  sin20. 


Area  of  Pedals  of  Ellipse  and  Hyperbola.  191 

Hence  its  area  is 

a2  f      o/iT/i      J2f   .  ,„,„      a2  +  b2n     a2-b2  .    n       a 

—    eotfOdO  +  —    sufOdd  = 0  +  sm0cos  0. 

2)2)  44 

The  entire  area  of  the  locus  is 

-  (a2  +  b2). 

2    X  ' 

The  equation  of  the  corresponding  locus  for  the  hyperbola 
is 

r2  =  a2cos20-62sin20. 

In  finding  its  area,  since  r  must  be  real,  we   must  have 
a2cos20  -  b2  sin2  0  positive :  accordingly,  the  limits  for  0  are  o 

and  tan-1 7. 
o 

Integrating  between  these  limits,  and  multiplying  by  4, 

we  get  for  the  entire  area 

ab  +  (a2  -  b2)  tan""1 7. 

In  this  case,  if  we  had  at  once  integrated  between  0  =  o 

and  0  =  27r,  we  should  have  found  for  the  area  (a2  -  b2)  -. 

2 

This   anomaly  would    arise    from   our  having    integrated 

through  an  interval  for  which  r2  is  negative,  and  for  which, 

therefore,  the  corresponding  part  of  the  curve  is  imaginary. 

The  expression  for  the  area  of  the  pedal  of  an  ellipse  with 

respect  to  any  origin  will  be  given  in  a  subsequent  Article. 

Examples. 

1.  Show  that  the  entire  area  of  the  Lemniscate 

r2  =  a~  cos  26 
is  a2. 

2.  In  the  hyperholic  spiral 

rO  =  a, 

prove  that  the  area  bounded  by  any  two  radii  veotores  is  proportional  to  the 
difference  between  their  lengths. 

3.  Find  the  area  of  a  loop  of  the  curve 

a2 
r2  =  a2  cos  n$.  Am.  — . 


192 


Examples, 


4.  Find  the  area  of  the  loop  of  the  Folium  of  Descartes,  whose  equation  is 

a3  +  y3  =  Zaxy. 

Transforming  to  polar  co-ordinates,  we  have 

3«  cos  0  sin  0 
r  =  V-r — . 

sin30  +  cos30 
Again,  the  limiting  values  of  0  are  o  and  - ; 


_  gar  fl  sin20cos20<f0 
ea==_2"J0  (sin30  +  cos30)3* 


Let  tan  0  =  u,  and  this  expression  becomes 


9«- 

2 


fw     u'du    _  id* 
Jo  (l+«3)3~T# 


5.  To  find  the  area  of  the  Limacon 

r  =  a  cos  0  +  b. 

Here  we  must  distinguish  between  two  cases. 

(1).  Let  b  >  a.     In  this  case  the  curve  consists  of  one  loop,  and  its  area  is 


1  f2»r  /  a2\ 

-I      (a  cos  0  +  b)*dd  =(&  +  -]  ir. 


lira? 
When  b  =  a,  the  curve  becomes  a  Cardioid,  and  the  area  - — . 

(2).  Let  b  <  a.  The  curve  in  this  case 
has  two  loops,  as  in  the  figure  (see  Diff. 
Calc,  Art.  269),  the  outer  loop  correspond- 
ing to 

r  =  a  cos  0  +  b, 
the  inner  to 

r  =  a  cos  0  —  b. 

To  find  the  area  of  the  inner  loop,  we      q 
take  0  between  the  limits  o  and  o,  where 

a  =  cos-1  ( -  j ;  and  the  entire  area  is 

[*  (acOBO  -b)2d0 
Jo 

=      (a2  cos20  -  iab  cos  0  +  &)  d0  Yi%.  13. 

(or     I9\  a?    . 

=  I  —  +  bl  J  a  +  —  sin  a  cos  a  -  2«£  sin  a 


Area  of  a  Closed  Curve  by  Polar  Co-ordinates.         193 


It  is  easily  seen  that  the  sum  of  the  areas  of  the  two  loops  is  obtained  hy  in- 
tegrating between  the  limits  o  and  2tt,  and  accordingly  is 


(?-)• 


as  in  the  former  case. 


137.  Area  of  a  Closed  Curve  by  Polar  Co-ordi- 
nates.— In  finding  the  whole  area  of  a  closed  curve  by- 
polar  co-ordinates  we  distinguish  between  two  cases.  When 
the  origin  0  is  outside,  we  sup- 
pose tangents  OT,  OT',  drawn 
from  0,  and  vectors  OP,  OQ,  &c, 
drawn  to  cut  the  curve ;  then,  if 
these  lines  intersect  it  in  but  two 
points  each,  the  element  of  area 
PpqQ  is  the  difference  between 
the  areas  POQ  and  pOq ;  or,  in 
the  limit,  is  %  (r2  -  r22)  dO,  where 
OP  =  rl9  Op  =  r2. 

Hence,  the  expression 


Fig.  14. 


taken  between  the  limits  corresponding  to  the  tangents  OT 
and  OT%  represents  the  entire  included  area. 

If  the  origin  lie  inside  the  curve,  its  whole  area  is  in  ge- 
neral represented  by  H(r2  +  r22)dd,  taken  between  the  limits 
0=o,  and  0  =  7r. 

"We  shall  illustrate  these  results  by  applying  them  to  the 
circle 

r2  -  zrc  cos  0  +  c2  =  a2. 

If  the  origin  be  outside,  we  have  e>a,  and  rx  +  r%  =  2c  cos  0, 

and  rxr2  =  c2  -  a2\  .  * .  rx  -  r2  =  2 ^/a2  -  c2  sin2 0. 

Hence  (n2  -  r22)  dO  =  4c  cos  0  */a2  -  &  sin2  0^0 ;  and  the 

limiting  values  of  0  are  ±  sin-1-. 

Hence  the  whole  area  is 


2C 


J  sin-1? 


cos  0  */a2 

fl3] 


(?am°9dd. 


104  Areas  of  Plane  Curves. 

Let  c  sin  0  =  a  sin  0,  and  this  integral  transforms  into 


2«2     2  cob2  <j>d<f>  =  na2. 


Again,  if  the  origin  be  inside,  we  have  c  <  a,  and 
-  (n2  +  r22)  =  a2  +  c2  cos  20 ; 

.*.  [Vi2  +  r22)  tf0  =  ["(a2  +  c2  cos  26)d0  =  na\ 

The  method  given  above  may  be  applied  to  find  the  area 
included  between  two  branches  of  the  same  spiral  curve.  As 
an  example,  let  us  consider  the  spiral  of  Archimedes. 

138.  The  Spiral  of  Archimedes. — The  equation  of 
this  curve  is  r  =  ad, 
and  its  form,  for 
positive*  values  of  0, 
is  represented  in 
the  accompanying 
figure,  in  which  0 
is  the  pole  and  OA 
the  line  from  which 
0  is  measured.  Let 
any  line  drawn 
through  0  meet  the 
different  branches 
of  the  spiral  in 
points  P,  Q,  R,  &c. : 
then,  if  OP=r,  and 
LP  OA  =  0,  we  have, 
from  the  equation 
of  the  curve, 

OP  =  aO,     OQ  =  a(0  +  2tt),     OR  =  a (0  +  4tt),  &c. 


*  It  should  be  noted  that  when  negative  values  of  6  are  taken,  we  get  for 
the  remaining  half  of  the  spiral  a  curve  symmetrically  situated  with  respect  to 
the  prime  vector  OA. 


Areas  by  Polar  Co-ordinates.  195 

Hence  PQ  =  QR  =  &c.,  =  zarc  =  c  (suppose) ;  i.  e.  the 
intercepts  between  any  two  consecutive  branches  of  the  spiral 
are  of  constant  length. 

Again,  let  OQ  =  ru  OR  =  r2  =  rx  +  c,  and  the  area  between 
the  two  corresponding  branches  is 


(r22  -  n2) dd  m  c  UdQ  +  -[dS. 


Now,  suppose  MN  and  mn  represent  the  limiting  lines, 
and  let  |3  and  a  be  the  corresponding  values  of  0 ;  then  the 
area  nNMm  will  be  equal  to 

c\\edB  +  -\Pd9  =  -(f5  -  a)  {aa  +  a[5  +  c) 

=  C-(P-a)(OM+On).  (9) 

If  (5  -  a  =  7r,  this  gives  for  the  area  of  the  portion 
between  two  consecutive  branches  QJE'Q'  and  MF'B',  inter- 
cepted by  any  right  line  RR'  drawn    through  the  pole, 

-RQ.QR',  i.e.  half  the  area  of  the  ellipse  whose  semi-axes 

are  RQ  and  R'Q. 

139.  Another  Expression  for  Area. — The  formula 
in  Article  137  still  holds,  obviously,  when  AB  and  a b  repre- 
sent portions  of  different  curves. 

It  is  also  easily  seen,  as  in  Art.  132,  that  if  a  point  be 
supposed  to  move  round  any  closed  boundary,  the  included 

area  is  in  all  cases  represented  by  -   r2dd,  taken  round  the 

entire  boundary,  whatever  be  its  form ;  the  elementary  angle 
dd  being  taken  with  its  proper  sign  throughout. 

Again,  if  we  transform  to  rectangular  axes  by  the  rela- 
tions x  =  r  cos  9,  y  =  r  sin  6,  we  get 

X  cos  20  X2 

Hence  r2dd  =  xdy  -  ydx ; 

[13  a] 


196 


Areas  of  Plane  Curves. 


and  the  area  swept  out  by  the  radius  vector  is  represented  by 
the  integral 


-Vxdy-ydx), 


a  result  which  can  also  be 


Lambert's    Tlieo- 


taken  between  suitable  limits ; 
easily  arrived  at  geometrically. 

140.  Area   of  Elliptic  Sector 

rem. — It  is  of  importance  in 
Astronomy  to  be  able  to  express 
the  area  AFP  swept  out  by  the 
focal  radius  vector  of  an  ellipse. 
This  can  be  arrived  at  by  inte- 
gration from  the  polar  equation 
of  the  curve ;  it  is,  however,  a  l 
more  easily  obtained  geometri- 
cally. 

For,  if  the  ordinate  PN  be  produced  to  meet  the  auxiliary 
circle  in  Q,  we  have 


area  AFP  =  -  x  area  AFQ  =  -(ACQ-  CFQ) 


ab  .  .      .  . 

=  — (u  -  e  sinw),  (10) 

where  u  =  lACQ. 

By  aid  of  this  result,  the  area  of  any  elliptic  sector  can  be 
expressed  in  terms  of  the  focal  distances  of  its  extremities, 
and  of  the  chord  joining  them. 

For  (Fig.  17),  let  QFP  re- 
present the  sector,  and  let 
FP  =  p,FQ  =  P',PQ  =  $;  then, 
denoting  by  u  and  u'  the  eccen' 
trio  angles  corresponding  to 
P  and  Q,  the  area  of  the  sector 
QFP,  by  ( 1  o) ,  is  represented  by 

ab 


N  A 


Fig.  17. 


—  \u  -  u'  -  e(sinw  -flinw')f, 


Lambert's  Theorem.  197 

We  proceed  to  show  that  this  result  can  be  written  in 
the  form 

—  [6  -  6'  -  (sin  6  -  sin^')}.  (n) 

where  6  and  6'  are  given  by  the  equations 


sin-  = 


£=i  jp  +  p+s     gin£  =  i  jp  +  P 

2        2  \  a  '  2         2  \  « 


For,  assume  that  0  and  6r  are  determined  by  the  equations 
u  -  u'  =  6  -  <jS,  e  (sin  u  -  sin  e/)  =  sin  0  -  sin  <j>'.       (a) 
The  latter  gives 


.    u  -  u        u  +  u       .6-6        6  +  6 
e  sin cos =  sin  - — —  cos 


i      ii      £  U  +  u'  6+6' 

or  by  the  former,   e  cos =  cos — . 


2  2  2  2 

u  +  u' 

2  2 

Again,  since  the  co-ordinates  of  P  and  Q  are  a  cos  u, 
b  sin  w,  and  a  cos «/,  5  sin  u',  respectively,  we  have 

S2  =  a2  (cos  w  -  cos  u')7,  +  b2  (sin  u  -  sin  w')2 


.  2u-u'(  2  .    u  +  u'     ,2      ,w  +  w'\ 

4  sin2 a2  sin2 +  b2  cos2 

2     V  2  2     J 

u  -u' (  .u  +  u'\ 

act  sin2 i  -  e2  cos2 

2     V  2     J 


—   An* 


.0-0       •20  +  # 

=  4#2  sm2  - — -  sm2  ■*■ — -  ; 

2  2 

•\  8  =  20  sin  - — —  sin  - — -  =  a  (cos  ^'  -  cos  6).         (b) 

Again,  from  the  ellipse,  we  have 

p  =  a(i  -  ecosu),       p  =  a(i  -  ecosu'), 

,  ,  ,x  U  +  U  U   -  It 

,  .  p  +  p  =  2a  — ae  (cos  u  +  cos  u )  =  2a  —  2ae  cos cos 

2  2 

=  za-  2a  cos  - — *  cos  * — -  =  za  -  a  (cos  ^  +  cos  6f).         (c) 


198  Areas  of  Plane  Curves. 

Hence,  adding  and  subtracting  (b)  and  (c),  we  get 

p  +  p'  +  8        .  ,  .  .<(> 

- =  2  ( i  -  cos  6)  =  4  sin2  r, 

a  r/  2 

p  +  p'  -  S       ,  ,  A         .  2  <t>' 

- — =  2(i  -  cos  <b  )  =4  sin2  — , 

a  v  r '  2 

which  proves  the  theorem  in  question. 

Consequently,  the  area*  of  any  focal  sector  of  an  ellipse  can 
be  expressed  in  terms  of  the  focal  distances  of  its  extremities,  of 
the  chord  which  Joins  them,  and  of  the  axes  of  the  curve. 

141.  We  next  proceed  to  an  elementary  principle  which 
is  sometimes  useful  in  determining  areas,  viz. : — 

The  area  of  any  portion  of  the  curve  represented  by  the 
equation 


%■  ?) 


is  ab  times  the  area  of  the  corresponding  portion  of  the  curve 
F(x,y)  =c. 
This  result  is  obvious,  for  the  former  equation  is  trans- 

x  II 

formed  into  the  latter,  by  the  assumption  -  =  x'f  -  =  7/ ;  and 
hence  ydx  becomes  aby'dx' ; 

.*.    ydx  =  ab    y'dx\ 

the  integrals  being  taken  through  corresponding  limits — a 

result  which  is  also  easily  shown  by  projection. 

x       u^ 
Thus,  for  example,  the  area  of  the  ellipse  -?  +  77  =  1 

a*      0 


*  This  remarkable  result  is  an  extension,  by  Lambert  (in  bis  treatise  entitled : 
Insigniorts  orbita  cometarum  proprieties,  published  in  1761),  of  the  correspond- 
ing formula  for  a  parabola  given  by  Euler  in  Mis  cell.  Berolin,  1743.  It 
furnishes  an  expression  for  the  time  of  describing  any  arc  of  a  planet's  orbit,  in 
terms  of  its  chord,  the  distances  of  its  extremities  from  the  sun,  and  the  major 
axis  of  the  orbit ;  neglecting  the  disturbing  action  of  the  other  bodies  of  the 
solar  system. 


Area  of  a  Pedal  Curve.  199 

reduces  to  that  of  the  circle  ;  and  the  area  of  the  hyperbola 

a*      V 

to  that  of  the  equilateral  hyperbola  #2  -  y%  =  i . 

Again,  let  it  be  proposed  to  find  the  area  of  the  curve 

^     f\~_  as*      f 
a1  +  ¥]~P  +  m2* 


The  transformed  equation  is 


(*2  +  tfY  =  -jr  +   A 


or,  in  polai  co-ordinates, 


2     a2cos70     ft2  sin2  0 
r  ~       P       +     m%     ' 

But  the  whole  area  of  this  (Art.  136)  is  -  ( —  +  — 
Consequently  the  whole  area  of  the  proposed  curve  is 

2      \/2      my 
It  may  be  remarked  that  the  equations 

represent  similar  curves,  and  their  corresponding  linear 
dimensions  are  as  a  :  1.  Consequently  the  areas  of  similar 
curves  are  as  the  squares  of  their  dimensions;  as  is  also 
obvious  from  geometry. 

142.  Area  of  a  Pedal  Curve. — If  from   any  point 
perpendiculars  be  drawn  to  the  tangents  to  any  curve,  the 


200  Area*  of  Plane  Curves. 

locus  of  their  feet  is  a  new  curve,  called  the  pedal  of  the 
original  (Diff.  Calc,  Art.  187). 

If  p  and  a)  be  the  polar  co- 
ordinates of  N,  the  foot  of  the 
perpendicular  from  the  origin  0, 
then  the  polar  element  of  area  of 
the  locus  described  by  iV  is  plainly  R/ 

- — ,  and  the  sectorial  area  of  any 

portion  is  accordingly  represented  by 


-  p2dto, 


taken  between  proper  limits. 

There  is  another  expression  for  the  area  of  a  closed  pedal 
curve  which  is  sometimes  useful. 

Let  81  denote  the  whole  area  of  the  pedal,  and  8  that  of 
the  original  curve ;  then  the  area  included  between  the  two 
curves  is  ultimately  equal  to  the  sum  of  the  elements  repre- 
sented by  NTN'  in  the  figure. 

Hence        8X  -  8  +  SiVTiT  =  8  4-  l-  [pN%d».  (12) 

Again,  by  the  preceding, 

8l=l-  \0Nzdto. 
Accordingly,  by  addition, 

COP2dto.  (13) 


28x  =  8  +  - 
2 


It  is  easily  seen  that  equation  (12)  admits  of  being  stated 
in  the  following  form : — 

The  whole  area  of  the  pedal  of  any  closed  curve  is  equal  to 
the  sum  of  the  areas  of  the  curve  and  of  the  pedal  of  its  evolute  : 
both  pedals  having  the  same  origin. 

For,  PN  is  equal  in  length  to  the  perpendicular  from  0 

on  the  normal  at  P :  and  hence  -PN^dta  represents  the  ele- 


Steiner's  Theorem  on  Areas  of  Pedal  Curves. 


201 


ment  of  area  of  the  locus  described  by  the  foot  of  this  perpen- 
dicular, i.e.  of  the  pedal  of  the  evolute  of  the  original  curve. 
For  example,  it  follows  from  Art.  136  that  the  area  0/ 

the  pedal  of  the  evolute  of  an  ellipse  is  -  (a  -  b)2,   the  centre 

being  origin. 

143.  Area  of  Pedal  of  Ellipse  for  any  Origin. — 

Suppose  0  to  be  the  pedal 
origin,  and  OM,  OM'  perpen- 
diculars on  two  parallel  tan- 
gents to  the  ellipse ;  draw  ON 
the  perpendicular  from  the 
centre  C;  let  OM  =  ply  OM' 
=  p2,  CJST  =  p,  OC  =c,  LOG  A 
=  a,  lACN  =  to;  then 


Pi  =  MB  -  OD  =  p  -  c  cos  (w 

pz  =  p  +  C  COS  (fa>  -  a) . 
Again,  the  whole  area  of  the  pedal  is 

-      {p2  +  p22)  da)  =\    [p2  +  C2  COS2  (w  -  a) )  dw 
2  J  0  Jo 


=     p2d(v  +  c2 


cos2 (w  -  a)du)  =  -(a2  +  b2  +  c2).      (14) 


That  is,  the  area  of  the  pedal  with  respect  to  0  as  origin 
exceeds  the  area  of  its  pedal  with  respect  to  0  by  half  the 
area  of  the  circle  whose  radius  is  OC. 

If  the  origin  0  lie  outside  the  ellipse,  the  pedal  consists 
of  two  loops  intersecting  at  0  and  lying  one  inside  the  other; 
and  in  that  case  the  expression  in  (14)  represents  the  sum  of 
the  areas  of  the  two  loops,  as  can  be  easily  seen. 

The  result  established  above  is  a  particular  case  of  a 
general  theorem  of  Steiner,  which  we  next  proceed  to 
consider. 

144.  Steiner's  Theorem  on  Areas  of  Pedal  Curves. 
Suppose  A  to  be  the  whole  area  of  the  pedal  of  any  closed 
curve  with  respect  to  any  internal  origin  0,  and  A'  the  area 


202  Areas  of  Plane  Curves. 

of  its  pedal  with  respect  to  another  origin  Of ;  then,  if  p  and 
p'  be  the  lengths  of  the  perpendiculars  from  0  and  (Zona 
tangent  to  the  curve,  we  have 

i  c2w  i  r2"" 

^  =  -     P*du,     A'  =  \\  P"d^ 

Also,  adopting  the  notation  of  the  last  article, 

p'  =  p  -  c  cos  (u)  -  a)  =  p  -  x  cos  u)  -  y  sin  to ; 

where  a?,  y  represent  the  co-ordinates  of  0'  with  respeot  to 
rectangular  axes  drawn  through  0.     Hence  we  get 

i  (2ir 
A!  -  A  =  -      (x  eos &  +  y  Bin  to)2 dit> 

2jo 

C2lT  T2JT 

-x\  pcoswdu)  -  y\    pBUKjjdu), 

r2jr  T27T   f  T27T 

But        cos2  w  du)  =  7r,       sin2  a>  G?u>  =  7r,       sinwcoso)  duj  =  o. 

Jo  Jo  Jo 

f2ir  T2ir 

Also,  for  a  given  curve,      ^?  coseu  da)  and      jt?  sinwdw  are 

Jo  J  0 

constants  when  0  is  given.  Denoting  their  values  by  g  and 
//,  we  have 

A'-A  =  ~(x2  +  y2)  -gx-  hy.  (15) 

This  equation  shows  that  if  0  be  fixed,  the  locus  of  the 
origin  Of,  for  which  the  area  of  the  pedal  of  a  closed  curve  is 
constant,  is  a  circle*  The  centre  of  this  circle  is  the  same, 
whatever  be  the  given  area,  and  all  the  circles  got  by  varying 
the  pedal  area  are  concentric. 


*  It  can  be  seen,  without  difficulty,  from  the  demonstration  given  above, 
that  when  the  curve  is  not  closed,  the  locus  of  the  origin  for  pedals  of  equal  area 
is  a  conic:  a  theorem  due  to  Prof.  Raabe,  of  Zurich.  See  Crelle's  Journal, 
vol.  1.,  p.  193. 

The  student  will  find  a  discussion  of  these  theorems  by  Prof.  Hirst  in  the 
Transactions  of  the  Royal  Society,  1863,  m  which  he  has  investigated  the  corre- 
sponding relations  connecting  the  volumes  of  the  pedals  of  surfaces. 


Areas  of  Roulettes. 


203 


If  the  origin  0  be  supposed  taken  at  the  centre  of  this 
circle,  the  constants  g  and  h  will  disappear ;  and,  in  this  case, 
the  pedal  area  is  a  minimum,  and  the  difference  between  the 
areas  of  the  pedals  is  equal  to  half  the  area  of  the  circle  whose 
radius  is  the  distance  between  the  pedal  origins. 

For  example,  if  we  take  the  origin  at  the  centre,  the 
pedal  of  a  circle,  whose  radius  is  a,  is  the  circle  itself.  For 
any  other  origin  the  pedal  is  a  limacon;   hence  the  whole 

area  of  a  limacon  is  jr*  a2  +  -  J,  as  found  in  Art.  136,  Ex.  5. 

145.  Areas   of  Roulettes  on   Rectilinear   Bases. 

The  connexion  between  the  areas  of  roulettes  and  of  pedals 
is  contained  in  a  very  elegant  theorem,*  also  due  to  Steiner, 
which  may  be  stated  as  follows  : — 

When  a  closed  curve  rolls  on  a  right  line,  the  area  between 
the  right  line  and  the  roulette  generated  in  a  complete  revolution 
by  any  point  invariably  connected  with  the  rolling  curve  is  double 
the  area  of  the  pedal  of  the  rolling  curve,  this  pedal  being  taken 
with  respect  to  the  generating  point  as  origin. 

To  prove  this,  suppose  0  to  be  the  describing  point  in  any 


Fig  20. 


position  of  the  rolling  curve,  and  P  the  corresponding  point 
of  contact.  Let  (J  represent  an  infinitely  near  position  of  the 
describing  point,  Q'  the  corresponding  point  of  contact,  and  Q 


*  See  Crelle's  Journal,  vol.  xxi.  The  corresponding  theorem  of  Steiner 
connecting  the  lengths  of  roulettes  and  pedals  will  be  given  in  the  next  Chapter. 

By  the  area  of  a  roulette  we  understand  the  area  between  the  roulette,  the 
base,  and  the  normals  drawn  at  the  extremities  of  one  segment  of  the  roulette. 


204  Areas  of  Plane  Curves. 

a  point  on  the  curve  such  that  PQ  =  PQ' ;  then  Q  is  the  point 
whioh  coincides  with  Q'  in  the  new  position  of  the  rolling 
curve ;  and,  denoting  the  angle  "between  the  tangents  at  P 
and  Q  (the  angle  of  contingence)  by  r/w,  we  have  OPO'  =  du), 
since  we  may  regard  the  curve  as  turning  round  P  at  the  in- 
stant (Diff.  Calc,  Art.  275). 

Moreover,  QQ'  ultimately  is  infinitely  small  in  comparison 
with  QP,  and  consequently  the  elementary  area  OPQ'O  is 
ultimately  the  sum  of  the  areas  POO  and  QO'P,  neglecting 
an  area  which  is  infinitely  small  in  comparison  with  either  of 
these  areas. 

Again,  if  OP  =  r,  we  have  POO '  = ,  and  area  QO'P 

=  QOP  in  the  limit. 

Also  the  sum  of  the  elements  QOP  in  an  entire  revolu- 
tion is  equal  to  the  area  (S)  of  the  rolling  curve.  Conse- 
quently the  entire  area  of  the  roulette  described  by  0  is 

S  +  iJr2du>. 

But  we  have  already  seen  (13)  that  this  is  double  the  area  of 
the  pedal  of  the  curve  with  respect  to  the  point  0 ;  which 
establishes  our  proposition. 

Again,  from  Art.  1 44,  it  follows  that  there  is  one  point  in 
any  closed  curve  for  which  the  entire  area  of  the  correspond- 
ing roulette  is  a  minimum.  Also,  the  area  of  the  roulette 
described  by  any  other  point  exceeds  that  of  the  minimum 
roulette  by  the  area  of  the  circle  whose  radius  is  the  distance 
between  the  points. 

For  instance,  if  a  circle  roll  on  a  right  line,  its  centre  de- 
scribes a  parallel  line,  and  the  area  between  these  lines  after 
a  complete  revolution  is  equal  to  the  rectangle  under  the 
radius  of  the  circle  and  its  circumference ;  i.e.  is  27m2 ;  denot- 
ing the  radius  by  a. 

Consequently,  for  a  point  on  the  circumference,  the  area 
generated  is  2-ira2  +  na2,  or  lira2 ;  which  agrees  with  the  area 
found  already  for  the  cycloid. 

In  like  manner,  by  Steiner's  theorem,  the  area  of  the  or- 
dinary cycloid  is  the  same  as  that  of  the  cardioid :  and  the 
area  of  a  prolate  or  curtate  cycloid  the  same  as  that  of  a 
limacon. 


General  Case  of  Area  of  Roulette.  205 

Again,  if  an  ellipse  roll  on  a  right  line,  the  area  of  the 
path  described  by  any  point  can  be  immediately  obtained. 

For  example,  the  pedal  of  an  ellipse  with  respect  to  a  focus 
is  the  circle  described  on  its  axis  major.  Hence,  if  an  ellipse 
roll  upon  a  right  line,  the  area  of  the  roulette  described  by  its 
focus  in  a  complete  revolution  is  double  the  area  of  the  auxiliary 
circle.  Also,  the  area  of  the  roulette  described  by  the  centre 
of  the  ellipse  is  equal  to  the  sum  of  the  circles  described  on 
the  axes  of  the  ellipse  as  diameters,  and  is  less  than  the  area 
of  the  roulette  described  by  any  other  point. 

146.  General  Case  of  Area  of  Roulette. — If  the 
curve,  instead  of  rolling  on  a  right  line,  roll  on  another 
curve,  it  is  easily  seen  that  the  method  of  proof  given  in  the 
last  article  still  holds ;  provided  we  take,  instead  of  du,  the 
sum  of  the  angles  of  contingence  of  the  two  curves  at  the 
point  P. 

Hence  the  element  of  area  OPO'  is  in  this  case 

-  OF>dw  (1  +  —\  or  -  OP*du  (l+A 

where  p  and  p'  are  the  radii  of  curvature  at  P  of  the  rolling 
and  fixed  curves,  respectively. 

Hence  it  follows  that  the  area  between  the  roulette,  the 
fixed  curve,  and  the  two  extreme  normals,  after  a  complete 
revolution,  is  represented  by 


'*?&*('.*$} 


If  a  closed  curve  roll  on  a  curve  identical  with  itself, 
having  corresponding  points  always  in  contact,  the  formula 
for  the  area  generated  becomes 

S  +  jr2doj. 

In  this  case  the  area  generated  is  four  times  that  of  the 
corresponding  pedal ;  a  result  which  appears  at  once  geome- 
trically by  drawing  a  figure. 


206 


Areas  of  Plane  Curves. 


Examples. 

i.  If  A  be  the  area  of  a  loop  of  the  curve  r»»  =  am  cos  m0,  and  A\  the  area 
of  its  pedal  with  respect  to  the  polar  origin,  prove  that 


-(,♦-)* 


It  is  easily  seen,  as  in  Diff.  Calc,  Art.  190,  that  the  angle  between  the  radius 
vector  and  the  perpendicular  on  the  tangent  is  md  ;  and  .*.  w  =  (m  +  i)d 
Hence,  by  Art.  142, 


2^!  =  ^+— 'jW* 


[m  +  2)A. 


2.  If  a  circle  of  radius  b  roll  on  a  circle  of  radius  a,  and  if  A  denote  the 
area,  after  a  complete  revolution,  between  the  fixed  circle,  the  roulette  described 
by  any  point,  and  the  extreme  normals ;  and  if  A'  be  the  area  of  the  pedal  of 
the  circle  with  respect  to  the  generating  point,  prove  that 

Aa  +  Bb  =  2(a  +  b)A'. 

where  B  is  the  area  of  the  rolling  circle. 

3.  Apply  this  result  to  find  the  area  included  between  the  fixed  circle  and  the 
arc  of  an  epicycloid  extending  from  one  cusp  to  the  next. 


147-  Holditch's  Theorem.*— If  a  line  CC  of  a  given 
length  move  with  its  extre- 
mities on  two  fixed  closed 
curves,  to  find,  in  terms  of 
the  areas  of  the  two  fixed 
curves,  an  expression  for  the 
whole  area  of  the  curve  gene- 
rated, in  a  complete  revolu- 
tion, by  any  given  point  P 
situated  on  the  moving  line. 

Let  CP  =  c,  PC  =  c\  and  suppose  (xh  yO,  (%,  ?/),  and 
(#2, 1/2)  to  be  the  co-ordinates  of  the  points  C,  P,  and  G\  re- 
spectively, with  reference  to  any  rectangular  axes. 


Fig.  21. 


f  This  simple  and  elegant  theorem  appeared,  in  a  modified  form,  as  the 
Prize  Question,  by  Mr.  Holditch,  under  the  name  of  "Petrarch,"  in  the  Lady's 
and  Gentleman's  Diary  for  the  year  1 858.  The  first  proof  given  above  is  due  to 
Mr.  "Woolhouse,  and  contains  his  extension  of  Mr.  Holditch' s  theorem. 


Holditcti's  Theorem,  207 

Then,  if  9  be  the  angle  made  by  CCf  with  the  axis  of  y, 
we  have  evidently 

xx  =  x  -  c  sin  0,    yx  =  y  -  c  cos  9, 

x2  =  x  +  c  sin  0,    y2  =  y  +  cf  cos  9. 

Hence  we  have 

yidxL  =  ydx  -  c  cos  9 (dx  +  yd9)  +  c2  cos2  0^0  ; 

y2dx2  =  ydx  +  c'  cos  9  (dx  +  yd9)  +  c"1  cos2 9 d9. 

Multiplying  the  former  equation  by  c',  and  the  latter  by  <?, 
and  adding,  we  get 

c'yidxi  +  cy2dx2  =  (c  +  c)  ydx  +  (c  +  c')  cc'  cos2  9d9 ; 

..'.  cf jyidxi  +  cjy2dx2  =  (c  +  c')\ydx  +  (c  +  c')cc' jcos?9d9. 

If  we  suppose  the  rod  to  make  a  complete  revolution,  so 
as  to  return  to  its  original  position,  and  if  we  denote  by  (0), 
(C),  (P),  the  areas  of  the  curves  described  by  the  points 
C,  C,  and  P,  respectively,  we  shall  have  (since  in  this  case 
the  angle  9  revolves  through  2tt) 

c\C)  +c(C)  -  (&+  c')(P)  +  tt(c  +  c')cc\ 

l^±^)=(P)+^,  (.6) 

This  determines  the  area  (P)  in  terms  of  the  areas  (0), 
(C)  and  of  the  segments  c,  c'. 

"When  the  extremities  C,  C  move  on  the  same  identical 
curve  we  have  (C)  =  (C),  and  hence  (C)  -  (P)  ■  ircc'. 

Consequently,  if  a  chord  of  given  length  move  inside  any 
closed  curve,  having  a  tracing  point  P  at  the  distances  c  and 
cf  from  its  ends,  the  area  comprised  between  the  two  curves  is 
equal  to  irccf. 

More  generally,  if  the  extremities  C,  C  move  on  curves 
of  equal  area,  we  have,  as  before, 

(C)-(P)=^'.  (17) 

Should  the  extremities,  instead  of  revolving,  oscillate 
back  to  their  former  positions,  then  (C)  =  o,  (C)  =  o,  and 


208  Areas  of  Plane  Curves. 

.*.  (P)  =  -  ircc.  The  negative  sign  implies  that  the  area  is 
described  in  a  direction  contrary  to  that  in  which  the  rod  re- 
volves. 

Again,  if  the  rod  returns  to  its  original  position  after 
n  revolutions,  the  limits  for  6  become  o  and  2inr,  and  equa- 
tion (i  6)  becomes 

'e  +  J    ;~(P)+W.  (18) 

If  (0)  =  (C),  this  gives 

(C)  -  (P)  =  mrcc'.  (19) 

If  the  line  oscillate  back  to  its  former  position,  without 
making  a  revolution,  we  have  n  =  o,  and  ( 1 9)  becomes 

(O)  =  (P). 

Hence,  in  this  case,  if  two  points  describe  curves  of  equal 
area,  then  any  point  on  the  line  joining  these  points  describes 
a  curve  of  the  same  area. 

The  theorem  in  ( 1 6)  can  also  be  proved  simply  in  another 
manner,  as  follows : — 

Let  0  denote  the  point  of  intersection  of  the  moving  line 
CC  with  its  infinitely  near  position ;  that  is  to  say,  the  point 
of  contact  with  its  envelope ;  and  let  OP  =  r.  Adopting  the 
same  notation  as  before,  let  ( 0)  represent  the  area  of  the  en- 
velope, and  it  is  easily  seen  that 

(c)  -  (oj-f  noo)'«»4 1*  («-♦■)««», 

Jo  Jo 

(C)  -  (0)  -t  J7oor*-i|j<+.f)*4 

(P)  -(0)-±j"(Oi')M»-ij!*<»; 
hence 

r2ir 

c\C)+c(Cf)-{c  +  c'){P)=%\     {c'ic-ry+cic'+rY-ic+c'j^cie 
=  cd  (c  +  C*)  IT, 

as  before. 


HolditcKs  Theorem.  209 

A  remarkable  extension  of  Holditch's  theorem  was  given 
by  Mr.  E.  B.  Elliott,  in  the  Messenger  of  Mathematics, 
February,  1878. 

Mr.  Elliott  supposed  the  length  of  the  moving  line  C'C  to 
vary,  but  that  it  is  in  all  positions  divided  in  the  constant 
ratio  m  :  n  in  a  point  P. 

Then,  if  C  travel  round  the  perimeter  of  any  closed  area 
(C),  and  C  move  simultaneously  round  another  area  {C%  the 
two  motions  being  quite  independent  and  subject  to  no  re- 
strictions whatever,  except  that  both  are  continuous,  having 
no  abrupt  passage  from  one  position  to  another  finitely  differ- 
ing from  it,  then  P  will  travel  simultaneously  round  the 
perimeter  of  another  closed  area  (P). 

Adopting  the  same  notation  as  before,  we  have 

(m  +  n)  x  =  mxx  +  nx2,     (m+n)y  =  myx  +  ny2 ; 

/.  (m  +  nYydx  =  (myx  +  ny2){mdxx  +  ndx2) 

=  m2yidxi  +  nzy2dx2  +  mn  {y2dxx  +  yxdx2) 

=  (m  +  n)  (myx  dxx  +  ny2  dx2)  -  mn  (y2  -  yj  d  [xt  -  xx) . 

Integrating  for  a  complete  circuit,  and  dividing  by  (m  +  n), 
we  have 

(m  +  n)(P)=m(C)  +  n(C')  -  -^-{(^-rid^-xt).     (20) 

lib  +  It  J 

This  result  is  stated  as  follows  by  Mr.  Elliott : — 
Through  any  fixed  point  in  the  plane  of  a  closed  area  S 
let  radii  vectores  be  drawn  to  all  points  in  its  perimeter,  and  let 
chords  AB,  parallel  and  equal  to  the  radii  vectores,  be  placed 
with  one  extremity  A  in  each  case  in  the  perimeter  of  a  closed 
area  (-4),  and  the  other  B  on  that  of  another  (B) ;  then,  if 
the  points  A,  P,  travel  respectively  all  round  the  perimeters, 
and  do  not  in  either  case  return  to  their  first  positions  from 
the  same  sides  as  that  towards  which  they  left  them ;  and,  if 

[14] 


210  Areas  of  Plane  Curves. 

(C)  represent  the  area  described  by  a  point  always  dividing  BA 
in  the  constant  ratio  m  :  n>  then  the  areas  (A),  (P),  (C),  (S) 
are  connected  by  the  following  relation  : 

{C)JnU)  +  n{B)_rnn 

m  +  n  (m  +  n)z  v    '  v     ' 

This  follows  immediately  from  (20)  by  altering  the  nota- 
tion. 

Areas  described  in  opposite  directions  of  rotation  must  be 
taken  with  opposite  signs. 

For  particular  modifications  in  this  result,  as  also  for  its 
extension  to  surfaces,  the  student  is  referred  to  Mr.  Elliott's 
paper  ;  as  also  to  Mr.  Leudesdorf's  papers  in  the  same 
Journal. 

147  (a),  liempe's  Theorem. — We  next  proceed  to  the 
consideration  of  a  singularly  elegant  theorem*  discovered  by 
Mr.  Kempe,  and  which  may  be  stated  as  follows : — 

If  one  plane  sliding  upon  another  start  from  any  position, 
move  in  any  manner,  and  return  to  its  original  position  after 
making  one  or  more  complete  revolutions ;  then  every  point 
in  the  moving  area  describes  a  closed  curve,  and  the  locus,  in 
the  moving  plane,  of  points  which  describe  equal  areas  is  a  circle ; 
and  by  varying  the  area  we  get  a  system  of  concentric  circles  for 
loci. 

This  result  can  be  readily  de- 
duced from  Holditch's  theorem,  for 
if  we  suppose  A,  B,  C,  to  be  three 
points  which  generate  equal  areas;  it 
can  easily  be  seen  that  any  fourth 
point,  D,  which  generates  the  same 
area,  lies  on  the  circle  circum- 
scribing ABC. 

Let  AB  and  CD  intersect  in  P, 
then,  let  (P)  represent  the  area 
described  by  the  point  P,  as  before ;  lg*  22' 

and  n  the  number  of  revolutions  made  before  AB  returns 
to  its  original  position :  then  we  have,  by  (19),  denoting  by 


Messenger  of  Mathematics,  July,  1878. 


Kemve's  Theorem.  211 

(C)    the   common   area   described  by  each  of    the   points 
A,  B,  G,  D, 

(C)-(P)=mrAP.PB, 

and,  by  same  theorem, 

(C)  -  (P)  =  nwCP.PD; 
hence 

AP.PB  =  CP.PB; 

consequently  A,  B,  C,  D,  lie  on  the  circumference  of  the 
same  circle. 

Again,  let  0  be  the  centre  of  this  circle,  and  join  OP  and 
OA,  then  the  preceding  equation  gives 

(C)  -  (P)  =  *w{QA%-  OP). 

Hence  all  points  which  describe  an  area  equal  to  that  of 
(P)  lie  on  a  circle,  having  0  for  centre,  and  OP  for  radius, 
which  establishes  the  second  part  of  the  theorem. 

For  the  effect  of  two  or  more  loops  in  the  area  described 
by  a  moving  point  see  Art.  132. 

148.  Areas  by  Approximation.- — In  many  cases  it  is 
necessary  to  approximate  to  the  value  of  the  area  included 
within  a  closed  contour.  The  usual  method  is  by  drawing  a 
convenient  number  of  parallel  ordinates  at  equal  intervals  ; 
then,  when  a  rough  approximation  is  sufficient,  we  may 
regard  the  area  of  the  curve  as  that  of  the  polygon  got  by 
joining  the  points  of  intersection  of  the  parallel  ordinates 
with  the  curve.  Hence,  if  h  be  the  common  distance  between 
the  ordinates,  and  if 

Vo,  t/i,  y*  &c.,  yn, 

represent  the  system  of  parallel  ordinates,  the  area  of  the 
polygon,  since  it  consists  of  a  number  of  trapeziums  of  equal 
breadth,  is  plainly  represented  by 


[14  a] 


212  Areas  of  Plane  Curves. 

Hence  the  rule :  add  together  the  halves  of  the  extreme 
ordinateSy  and  the  ichole  of  the  intermediate  ordinates,  and 
multiply  the  result  by  the  common  interval. 

When  a  nearer  approximation  is  required,  the  method 
next  in  simplicity  supposes  the  curve  to  consist  of  a  number 
of  parabolic  arcs ;  each  parabola  having  its  axis  parallel  to 
the  equidistant  ordinates,  and  being  determined  by  three  of 
those  ordinates. 

To  find  the  area  of  the  parabola  passing  through  the 
points  whose  ordinates  are  y0,  yXi  y* ;  let  y  =  a  +  fix  +  yx1  be 
the  equation  of  the  parabola,  and,  for  simplicity,  assume  the 
origin  at  the  foot  of  the  intermediate  ordinate  yx,  then  we 
have 

yQ  =  a-  fih  +  yh2,      yy  m  a,      y2  =  a  +  fih  +  yK\ 

Again,  the  area  between  the  first  and  third  ordinate  is 

(a  +  fix  +  yx2)  dx  =  2h  ( a  +  y  —  J. 

But  yo  +  yz=  2yx  +  2yh2 :  hence  the  area  in  question  is 
hi 


-  Wo  +  42/i  +  y* 

Now,  if  we  suppose  the  number  of  intervals  n  to  be  even, 
and  add  the  different  parabolic  areas,  we  get,  as  an  approxi- 
mation to  the  area,  the  expression 

-  {2/o  +  2/«  +  4(yi  +  y3+&c.  +  2/n_1)  +  2(y2  +  y4  +  &c.+y„_2)). 
o 

Hence  the  rule  :  add  together  the  first  and  last  ordinates, 
twice  every  second  intermediate  ordinate,  and  four  times  each 
remaining  ordinate;  and  multiply  by  one-third  of  the  common 
interval. 

We  get  a  closer  approximation  by  supposing  the  number 
of  equal  intervals  a  multiple  of  3,  and  regarding  the  curve 
as  a  series  of  parabolse  of  the  third  degree,  each  being 
determined  by  four  equidistant  ordinates.  To  find  the  area 
corresponding  to  one  of  these  parabolic  curves,  let  y0>  jfa  y2,  ys 
be  four  equidistant  ordinates,  and  for  convenience  assume 


Areas  by  Approximation.  213 

the  origin  midway  between  yx  and  y2 ;  then  if  the  equation 
of  the  parabolic  curve  be 

y  =  a  +  fix  +  yx2  +  &c3, 
and  the  common  interval  on  the  axis  of  x  be  denoted  by  2  h, 
we  have 

yo  =  a-  3J5h+  gyh2-  278A3, 

y1  =  a  -  fih  +  yh2  -  U3, 
y2  ■  a  +  (3h  +  yh2  +  SA3, 
2/3  =  a  +  $fih  +  gyh2  +  27M3. 
Hence     y0  +  y3  =  2  (a  +  gyh2),     yx  +  y2  =  2{a  +  yh2). 
Again,  the  parabolic  area  between  y0  and  ys  is 

JZh 
(a  +  (3x  +  yx2  +  §x*)dx  =  Sh(2a  +  6yh2). 
-zh 

Substituting  in  this  the  values  of  a  and  7  obtained  from 
the  two  preceding  equations,  the  expression  for  the  area 
becomes 

V  {yo  +  y3  +  3(^1  +  2/2)}. 
4 

If  the  corresponding  expressions  be  added  together,  we 
easily  arrive  at  the  following  rule  :* — Add  together  the  first 
and  last  ordinates,  twice  every  third  intermediate  ordinate,  and 
thrice  each  remaining  ordinate ;  and  multiply  by  fths  of  the 
common  interval. 

It  is  readily  seen  that  these  rules  also  apply  to  the  ap- 
proximation to  any  closed  area,  by  drawing  a  system  of  lines, 
parallel  and  equidistant,  and  adopting  the  intercepts  made  by 
the  curve  instead  of  the  ordinates,  in  each  rule. 

Since  every  definite  integral  may  be  represented  by  a 

*  This  and  the  preceding  are  commonly  called  "  Simpson's  rules  "  for  cal- 
culating areas  ;  they  were  however  previously  noticed  hy  Newton  (see  Opuscula. 
Method.  Biff.,  Prop.  6,  scholium)  as  a  particular  application  of  the  method  of 
interpolation.  By  taking  seven  equidistant  ordinates,  Mr.  "Weddle  (Camb.  and 
Dub.  Math.  Jour.,  1854),  ohtained  the  following  simple  and  important  rule  for 
finding  the  area: — To  Jive  times  the  sum  of  the  even  ordinates  add  the  middle  ordi- 
nate and  all  the  odd  ordinates,  multiply  the  sum  by  three-tenths  of  the  common 
interval,  and  the  product  will  be  the  required  area,  approximately.  The  proof, 
which  is  too  long  for  insertion  here,  will  he  found  in  Mr.  "Weddle' s  memoir : 
and  also,  with  applications,  in  Boole's  Calculus  of  Finite  Differences.  The  student 
is  referred  to  Bertrand's  Gale.  Int.,  I.  1,  ch.  xii.,  for  more  general  and  accurate 
methods  of  approximation  hy  Cotes  and  Gauss. 


214 


Areas  of  Plane  Curves. 


curvilinear  area,  the  methods  given  above  are  applicable  to 
the  approximate  determination  of  any  such  integral. 

In  practice  the  accuracy  of  these  methods  is  increased  by 
increasing  the  number  of  intervals. 

149.  Planimeters. — Several  mechanical  contrivances 
have  been  introduced  for  the  purpose  of  practically  estimating 
the  area  inclosed  within  any  curved  boundary.  Such  instru- 
ments are  called  Planimeters.  The  simplest  and  most  elegant 
is  that  of  Professor  Amsler  of  Schaffhausen.  It  consists  of 
two  arms  jointed  together  so  as  to  move  in  perfect  freedom  in 
one  plane.  A  point  at  the  extremity  of  one  arm  is  made  a 
fixed  centre  round  which  the  instrument  turns ;  and  a  wheel 
is  fixed  to,  and  turns  on  the  other  arm  as  an  axis,  and  records 
by  its  revolution  the  area  of  the  figure  traced  out  by  a  point 
on  this  arm.  From  its  construction  it  is  plain  that  the  re- 
volving wheel  registers  only  the  motion  which  is  perpendi- 
cular to  the  moving  arm  on  which  it  revolves. 

In  the  practical  application  of  the  instrument  it  is  neces- 
sary that  the  two  arms,  CA  and  AB,  should  return  to  their 
original  position  after  the  tracing  point  B  has  been  moved 
round  the  entire  boundary  of  the  required  area. 

We  shall  commence  by  showing  that  the  length  registered 
by  the  wheel  while  B  has  moved  round  the  entire  closed  area 
is  independent  of  the  wheel's  position  on  the  moving  arm ; 
i.e.  is  the  same  as  if  the  wheel  be  supposed  placed  at  the  joint. 

To  prove  this,  suppose  P  to  represent  the  point  on  the 
arm  at  which  the  centre  of  the 
revolving  wheel  is  situated.  Let 
A'B'  represent  a  new  position  of 
AB  very  near  to  AB,  and  P'  the 
corresponding  position  of  the 
point  P.  Draw  PN perpendicular 
to  A' If  ;  then  PN  represents  the 
length  registered  by  the  wheel 
while  the  arm  moves  from  AB  to 
the  infinitely  near  position  AB ' . 

Next,  draw  AN  perpendicular, 
and  AL  parallel,  to  A'B". 

Let  PN=  ds\  AN'  =  ds,  AP  =  e, 
PAL  =  d+ ;  then  PN=  PL  +  AN', 
or  ds'  =  ds  +  c  dd>. 


Amsler's  Planimeter. 


215 


Now,  if  we  suppose  AB  after  a  complete  circuit  of  the 
curve  to  return  to  its  original  position,  we  have  obviously 
2  (dcp)  =  o  ;  and  therefore  2  (ds)  =  2  (d*),  i.e.  the  whole  length 
registered  by  the  revolving  wheel  at  P  is  the  same  as  if  it 
were  placed  at  A. 

Next,  let  x  and  y  be  the  co-ordinates  of  B  with  respect  to 
rectangular  axes  drawn  through  (7,  and  let  AC  =  a,  AB  =  b, 
L  ACX  =  9  ;  and  suppose  <p  the  angle  which  BA  produced 
makes  with  the  axis  of  x  ;  then  we  shall  have 

x  =  a  cos  9  +  b  cos  tf>,     y  =  a  sin  9  +  b  sin  0. 
Hence     xdy  -  ydx  =  a2d9  +  b2d<j>  +  ab  cos  (0  -  (p)  d{9  +  <f>). 
Also        ds  =  AN'  =  A  A!  sm.AA!N  =  ad9  cos  (9  -  </>). 
But  9  +  <f>  =  z9- {9-<t>); 

.-.  ab  coa  [9  -  (p)d(9  +  <[>) 
=  2ab  cos(0  -  (f)d9  -  ab  cos(0  -  <p)  d{9  -  <f>) 
=  zbds  -  ab  cos  (9  -  </>)  d{9  -  0). 
Consequently 

xdy-ydx  =  a?d9  +  b2d<j>  +  zbds  -  ab  cos(0-  <f)d(9  -  $). 
But,  by  Art.  139,  the  area  traced  out  by  B  in  a  complete 


revolution  is  represented  by  ■^ 


(xdy  -  ydx)  taken  around  the 


entire  curve. 

Also,  since  AC  and  AB  return  to  their  original  positions, 
the  integrals  of  the  terms  a2d9,  b2d<f>  and  ab  cos  (9  -  <j>)  d(9-  <j>) 
disappear  ;  and  hence  the  area  in  question  is  equal  to  bS,  where 
S  denotes  the  entire  length  registered  by  the  revolving  wheel. 

On  account  of  the  importance  ~of  the  principle  of  this  in- 
strument, the  following  proof,  for  b 
which  I  am  indebted  to  Prof.  Ball, 
based  on   elementary   geometrical 
principles,  is  also  added. 

Let  C,  A,  i?  represent,  as  before, 
the  positions  of  the  fixed  centre,  the 
joint,  and  the  tracing  point,  respec- 
tively ;  and  suppose  B  to  represent 
the  position  of  the  roller,  or  revolv- 
ing wheel ;  then  draw  CP  and  RS 
perpendicular  to  AB. 


216 


Areas  of  Plane  Curves. 


Let         AC  =  a,  AB  =  b,  AR  =  /,  BC  =  r. 

Now,  if  the  instrument  be  rotated  about  C  through  an 
angle  0  without  altering  the  angle  CAB,  it  is  easily  seen 
that  the  circumference  of  the  roller  is  rotated  through  an  arc 
represented  by 


PJR  .0=    1  + 


+  b2 


> 


Again,  if  the  instrument  be  rotated  about  S  through  a 
small  angle  the  roller  does  not  revolve. 
Hence  a  curve  can  be  drawn  through  B, 
such  that,  if  the  tracing  point  B  be 
moved  along  it,  the  roller  will  not 
revolve. 

Now,  let  X/x,  XV  De  the  two  adjacent 
circles  described  with  C  as  centre,  and 
suppose  aa  and  (S|3'  two  adjacent  non- 
rolling  curves,  such  as  just  stated :  and 
suppose  the  tracing  point  B  to  move 
round  the  indefinitely  small  area  aa'fifi  :  then  the  arc  through 
which  the  roller  has  turned  is  represented  by 


1  + 


a2  +  b2-  r2 

2b 


$#-(!+ 


a2  +  b2  -  (r  +  Sr) 

2b 


:) 


86 


=  — : —  =  area  of  — !rJ-, 
b  b 

since  a|3  =  r  $0 ;  and  Sr  =  aa  sin  )3. 

Now  suppose  the  instrument  works  correctly  for  the  area 
XXVa,  then  it  will  work  correctly  for  the  area  XX'j3'|3 ;  for, 
start  from  a  to  X,  X',  a',  then  the  area  aXXV  must  be  regis- 
tered, since  the  roller  does  not  turn  in  moving  from  a'  to  a ; 
proceed  then  from  a'  to  j3',  )3,  a,  then,  by  what  has  been  just 
proved,  the  area  a  'ft (3a  will  be  added.  Hence  the  instrument 
will  work  correctly  for  the  strip  XX'/u'/u* 

Again,  suppose  the  instrument  works  correctly  for  the 
area  Xjup,  then  it  will  work  correctly  for  X'fip  ;  for  suppose 
we  start  from  X  to  p,  /x,  and  back  to  X  :  then  start  from  X  to 


Amsler's  Planimeter.  217 

/u,  //,  X  and  X ;  the  two  journeys  from  X  to  fi  and  /u  to  X 
will  neutralize  each  other,  and  it  follows  that  if  the  instrument 
works  correctly  for  the  area  Ajup,  it  will  work  correctly  for 
the  area  Xfip  :  hence,  if  the  instrument  works  correctly  for 
any  portion  of  the  area,  however  small,  it  works  correctly  for 
the  entire  area. 

The  student  will  find  a  description  of  Amsler's  Planimeter, 
with  another  mode  of  demonstration,  in  a  communication  by 
Mr.  F.  J.  Bramwell,  O.E.,  to  the  British  Association. — See 
Eeport,  1872,  pp.  401-412. 


218  Examples. 

Examples, 
i.  Find  the  whole  area  hetween  the  curve 

x2y2  +  a2b2  =  a-y* 


and  its  asymptotes. 

Ans.  2irab. 

2.  Find  the  whole  area  of  the  curve 

aV  =  *4(«2-*2). 

"     s' 

3.  Find  the  whole  area  of  the  curve 

(;)'♦  (?)'-* 

„     \*ab. 
4 

4.  Find  the  whole  area  included  hetween  the  folium  of  Descartes 

s3  +  y3  -  z<*xy  =  0 

and  its  asymptote. 

An*.  >*. 

5.  In  the  logarithmic  curve  y  =  a',  prove  that  the  area  hetween  the  axis  of 
x  and  any  two  ordinates  is  proportional  to  the  difference  between  the  ordinates. 

6.  Find  the  area  of  a  loop  of  the  curve 

r  =  a  cos  nd.  Ans.  — . 

n 

7.  Find  the  area  of  a  loop  of  the  curve 

r  =  a  cos  n$  +  b  sin  nd.  „     (a2  +  b%)  -. 

The  equation  of  the  curve  may  he  written  in  the  form 

r  =  v^a2  +  b2  cos  (nd  +  o), 

where  tan  a  — ;  and  consequently  its  area  can  be  found  from  the  preceding; 

example. 

8.  Find  the  area  of  a  loop  of  the  curve 


r2  =  a2  cos  nd  +  b2  sin  nd.  Ans. 


/a*  +  b* 


Examples. 


219 


9.  Find  the  area  of  the  tractrix. 

The  characteristic  property  of  the  tractrix  is  that  the  intercept  on  a  tangent 
to  the  curve  between  its  point  of  contact  and  a  fixed  right  line  is  constant. 

Denoting  the  constant  by  a,  and  taking  the  origin  0  at  the  point  for  which 
the  tangent  OA  is  perpendicular 
to  the  axis,  we  have,   P  being 
any  point  on  the  curve 


FT- 


PN=yt 


.*.  ydx  =  -  \/a2  —  y2dy. 

Hence  the  element  of  the  area  of 
the  tractrix  is  equal  to  that  of 
a  circle  of  radius  a. 

It  follows  immediately  that  the  whole  area  between  the  four  infinite  branches 
of  the  tractrix  is  equal  to  ?ra2.  This  example  furnishes  an  instance  of  our  being 
able  to  determine  the  area  of  a  curve  from  a  geometrical  property  of  the  curve, 
without  a  previous  determination  of  its  equation. 

If  the  equation  of  the  tractrix  be  required,  it  can  be  derived  from  its  differ- 
ential equation 


dx  = 


from  which  we  get 


x  + 


V  a2 


*/a2  -  f-dy 

y       ' 

a  +  Cat- 
alog   

-  !/' 

That  the  equation  of  the  tractrix  depends  on  logarithms  was  noticed  by 
Newton.  See  his  Second  Epistle  to  Oldenburg  (Oct.  1676).  This  was,  I 
believe,  the  first  example  of  the  determination  of  the  equation  of  a  curve  by 
integration ;  or,  what  at  the  time  was  called  the  inverse  method  of  tangents. 

10.  If  each  focal  radius  vector  of  an  ellipse  be  produced  a  constant  length  c, 
show  that  the  area  between  the  curve  so  formed  and  the  ellipse  is  nrc{2b  +  c), 
b  being  the  semi-axis  minor  of  the  ellipse. 

11.  Find  the  area  of  a  loop  of  the  curve  r»  =  an  cos  n9. 


Ans. 


*v; 


m 


a 


BC x  (A)  +  CAx  (B)  +ABx(C)+ir.AB.  BC.  GA  m  o, 

BC,  &c,  are  taken  with  their  proper  signs  ; 


in  which  the  lines  AB 
AB  =  -  BA,  &c. 


12.  If  a  right  line  carrying  three  tracing  points  A,  B,  G,  move  in  any  manner 
in  a  plane,  returning  to  its  original  position  after  making  a  complete  revolution ; 
and  if  (A),  (B),  (C)  represent  the  entire  areas  of  the  closed  curves  described  by 
the  points  A,  B,  C,  respectively,  prove  that 


220  Examples. 

13.  A,  B,  C,  D,  are  four  points  rigidly  connected  together,  and  moving  in 
nny  way  in  a  plane  ;  if  they  describe  closed  curves,  of  areas  (A),  (B),  (C),  (2>), 
respectively ;  and  if  x,  y,  z,  he  the  areolar  co-ordinates  of  D  referred  to  the 
triangle  ABC,  prove  that 

(D)  =  x(A)  +  </{B)  +  z(C)-7rt\ 

where  t  is  the  length  of  the  tangent  from  D  to  the  circle  circumscribed  to  the 
triangle  ABC.     Mr.  Leudesdorf,  Messenger  of  Mathematics,  1878. 

This  follows  immediately :  for  let  P  be  the  point  of  intersection  of  the  lines 
AB  and  CD,  then,  by  (18),  we  get  a  relation  between  (A),  (B),  and  (P) ;  and 
also  between  (C),  (D),  and  (P).  If  P  be  eliminated  between  these  equations  we 
get  the  required  result. 

14.  Show  that  a  corresponding  equation  connects  the  areas  of  the  pedals  of 
any  given  closed  curve  with  respect  to  four  points  A,  B,  C,  JD,  taken  respectively 
as  pedal  origin.    Mr.  Leudesdorf. 

15.  If  a  curve  be  referred  to  its  radius  vector  r  and  the  perpendicular  p  on 
the  tangent,  prove  that  its  area  is  represented  by 


51 


prdr 


Si 


16.  A  chord  of  constant  length  (c)  moves  about  within  a  parabola,  and 

tangents  are  drawn  at  its  extremities ;  find  the  total  area  between  the  parabola 

and  the  locus  of  intersection  of  the  tangents. 

ire2 
Ans.  — . 

2 

17.  From  the  centre  of  an  ellipse  a  tangent  is  drawn  to  a  semicircle 
described  on  an  ordinate  to  the  axis  major ;  prove  that  the  polar  equation  of  the 
locus  of  the  point  of  contact  is 

a2*2 


b2  +  (a2  +  P)  tan2  6 
and  that  the  whole  area  of  the  locus  is 


2V/«2-r*2  +  * 

18.  Apply  the  three  methods  of  approximation  of  Art.  148  to  the  calculation 

to  6  decimal  places  of  the  definite  integral  I      ,  adopting  —  as  the  common 

Jo  1  +a?  12 

interval  in  each  case.  Ans.  (1),  .693669.     (2),  .693266.     (3),  .693224. 

The  'eal  value  of  the  integral  being  log  2,  or  .693147,  to  the  same  number 
of  decimal  places. 

1 9.  Prove  that  the  sectorial  area  bounded  by  two  focal  vectors  r  and  r'  of  a 
parabola  is  represented  by 

where  e  is  the  chord  of  the  arc,  and  a  the  semiparameter  of  the  parabola. 


Examples.  221 

20.  Show  that  the  whole  area  of  the  inverse  of  the  ellipse  —  +  —  =  i  is 

a*      bz 

represented  by 

irk*  (r       I       H_l\  (<>l_P\\ 

V       a*      b*] 

where  a,  £,  are  the  co-ordinates  of  the  origin  of  inversion,  and  k  is  the  radius  of 
the  circle  of  inversion. 

ai.  A  given  arc  of  a  plane  curve  turns  through  a  given  angle  round  a  fixed 
point  in  its  plane ;  what  is  the  area  described  ? 

22.  Given  the  base  of  a  triangle,  prove  that  the  polar  equation  of  the  locus 
of  its  vertex,  when  the  vertical  angle  is  double  one  of  its  base  angles  is 

a (2  cos  2d  +  i) 
r  =  - . 

2  COS  0 

Hence  show  that  the  entire  area  of  the  loop  of  the  curve  is  3a  V.  }t 

4 

23.  0  is  a  point  within  a  closed  oval  curve,  P  any  point  on  the  curve,  QPQ' 
a  straight  line  drawn  in  a  given  direction  such  that  QF  =  PQ'  =  PO  ;  prove  that 
as  P  moves  round  the  curve,  Q,  Q',  trace  out  two  closed  loops  the  sum  of  whose 
areas  is  twice  the  area  of  the  original  curve.     Camb.  Trip.  Exam.,  1874. 

24.  Prove  that  the  area  of  the  pedal  of  the  cardioid  r  =  a  (1  -  cos  6)  taken 
with  respect  to  an  internal  point  at  the  distance  c  from  the  pole  is 

^  (S«8  -  2ac  +  2c2).  (Ibid.,  1876.) 

8 

25.  The  co-ordinates  of  a  point  are  expressed  as  follows  : 
3^  303 


x  = 


03  +  l'        *         03  +  I  ' 


find  the  equation  of  the  curve  described  by  the  point,  and  the  area  of  the  portion 
of  the  plane  inclosed  thereby. 


(     222     ) 


CHAPTEE    VIII. 

LENGTHS   OF   CURVES. 

1 50.  Length  of  Curves  referred  to  Rectangular  Axes. 

The  usual  mode  of  considering  the  length  of  a  curve  is  by 
treating  it  as  the  limit  of  a  polygon  when  each  of  its  sides  is 
infinitely  small.  If  the  curve  be  referred  to  rectangular  axes 
of  co-ordinates,  the  length  of  the  chord  joining  the  points 
(x,  y)  and  (x  +  dx,y  +  dy)  is  */dx2  +  dy2,  and,  consequently,  if 
s  represent  the  length  of  the  curve  measured  from  a  fixed 
point  on  it,  we  shall  have  ds  =  */3&  +  dy2,  or,  integrating, 


Mi 


2 
dx,  ( 1 ) 

taken  between  suitable  limits. 

dii 
The  value  of  —  in  terms  of  x  is  to  be  got  from  the  equa- 
ctx 

tion  of  the  curve,  and  thus  the  finding  of  s  is  reducible  to  a 

question  of  integration. 

The  determination  of  the  length  of  an  arc  of  a  curve  is 
called  its  rectification. 

It  is  evident  that  if  y  be  taken  for  the  independent  variable 
we  shall  have 


■u 


^■{4)  dv- 


Again,  when  x  and  y  are  given  functions  of  a  single  va- 
riable 0,  we  have 


■mm* 


In  each  case  the  form  of  the  equation  of  the  curve  deter- 
mines which  of  these  formulae  should  be  employed. 


The  Catenary. 


223 


The  curves  whose  lengths  can  be  obtained  in  finite  terms 
(compare  Art.  2)  are  very  limited  in  number.  We  proceed  to 
consider  some  of  the  simplest  cases. 

151.  The    Parabola. — "Writing  the   equation   of  the 

dx      y 


parabola  in  the  form  y%  =  zmx,  we  get 


dy     m 


Hence 


vV 


+  m* 


The  value  of  this  integral  can  be  obtained  from  that  of 
the  area  of  a  hyperbola  (Art.  130),  by  substituting  y  for  x, 
and  m2  for  -  a2. 

Thus  we  have 

-%(—  ~^r — }  (2) 


s  = 


y</y* 


2m 


m 


the  arc  being  measured  from  the  vertex  of  the  curve. 

152.  The    Catenary. — The  equation  of  the   catenary 
(Art.  131),  is 


y  =  -(e*+  e~"\ 


Hence 


dy 
dx 


<h_  _  (       dyy 
dx  ~  \      dx2) 


eadx  +  - 
2 


e  a  dx  = 


If  s  be  measured  from  the  vertex  V,  we  have 
a 


the  same  result  as  already  arrived  at  in  Art.  1 31 . 

Again,  since  PL  =  P  V,  and  NL  is  constant,  it  follows  that 
the  catenary  is  the  evolute  of  the  tractrix  (see  Ex.  9,  p.  219). 


224  Lengths  of  Curves. 

153.  No  mi -cubit*  ill  Parabola. — The  equation  of  this 
curve  is  of  the  form  ay2  =  a?8. 

h  m** .  .  *a  m  ifftf  fa « (   ??Y- 

fli '  '  "  dx     2  \aj  '     dx     \       40/  ' 
.'.  s  =    (  1  +  —  )  dx  =  —  (  1  +  —  )  +  const. 

If  the  arc  be  measured  from  the  vertex,  we  get 

8a  (/       gx 
S~  27  l\       4«. 

The  semi-cubical  parabola  is  the  first  curve  whose  length 
was  determined.  This  result  was  discovered  by  William 
Neil,  in  1660. 

154.  Rectification  of  K volutes. — It  may  be  noted 
that  the  rectification  of  the  semi-cubical  parabola  is  an 
immediate  consequence  of  its  being  the  evolute  of  the  ordinary 
parabola  (see  Diff.  Calc,  Art.  239).  In  like  manner  the 
length  of  any  curve  can  be  found  if  it  be  the  evolute  of  a 
known  curve,  from  the  property  that  any  portion  of  the  arc 
of  the  evolute  is  the  difference  between  the  two  corresponding 
radii  of  curvature  of  the  curve  of  which  it  is  the  evolute. 

For,  example,  we  get  by  this  means  the  lengths  of  the 
cycloid,  the  epicycloid  and  the  hypocycloid. 

Again,  since  the  equation  of  the  evolute  of  an  ellipse  is 

(ax)%  +  {byfl  =  (a2  -  b% 

the  length  of  any  arc  of  this  curve  can  be  at  once  found. 

This  can  also  be  readily  got  otherwise ;  for,  writing  the 
equation  in  the  form 


© 


<$.„ 


and  making  x  =  a  sin3^,  we  get  y  =  (5  cos3tf>,  and 

ds  =  (dx2  +  dy2)^  =  3  sin  <j>  cos^)(a2  sin20  +  j32  co&2  <p)^d<p 

3(a2  siuty  +  j32cos2tf>)*    ,  ,    .  2j      02       .  x 
2(a2-/32)  (  0  +  ■*         ^' 


Examples  in  Rectification.  225 

Hence 

(a2sin2d>  + j32cos2d>)* 

s  m  ~ ,     a. —  +  const. 

a2  -  |32 

If  the  arc  be  measured  from  the  point  x  =  o,  y  =  /3,  we 
get  the  constant 

-  /33         ,        (a8  sin2  0  +  |32cos20)i-j33 


and  s  = 


-  j32'  a2  -  ]3S 


If  a  =  j3,  the  expression  for  ds  becomes  3a  sin  0  cos 
hence  we  get  s  =  -  a  sin2^>,  the  arc  being  measured  from  the 
same  point  as  above. 

Examples. 
1.  Find  the  length  of  the  logarithmic  curve  y  —  ca*. 

Here  log  y  =  x  log  a  +  log  c ;    .-.-—  =  -,  where  b  =  . . 

dy      y'  log  a 


{b*  +  y*)ldy  f  ydy 
(*2  +  y'' 
(J2  +  y2)J 


Hence  ,f(*2  +  ^=  f  _J^_.      f       ^dy 


=  (b*  +  y*)l  +  J  log 

y 
2«  Find  the  length  of  the  tractrix. 
Here,  by  definition  (see  fig.  26),  we  have  FT  =  a  ; 

.-.  sin  PTiV  =  J-.      hence  —  =  -  - ; 
a  dy         y 

.*.  *  =  -  a\  —  =  -  a  log  y  +  const. 

If  the  arc  be  measured  from  the  vertex  A,  we  get 

arc  AP  =  a  log  ( -  J . 

3.  Find  in  what  cases  the  curves  represented  by  amyn  =  xmJ"x  are  rectifiabld. 
Here  we  have 

f  (         /m  4-  n\  2  /#\ 

I 


[15] 


226  Lengths  of  Curves. 

(m  +  n)2  2™ 

Substituting  b  for — -,  and  making  i  +  bx  »  -  z2,  this  becomes 


_n_  f  lz2-  i\ 
~mb)  \     b     ) 


This  expression  is  immediately  integrable  when  —  is  a  positive  integer. 

Hence,  if  —  =  r,  we  see  that  curves  of  the  form  ay2r  =  x2r+l  are  rectifiable. 
2m 

Again,  if  —  be  a  negative  integer,  the  expression  under  the  integral  sign 

becomes  rational,  and  can  accordingly  be  integrated.  This  leads  to  the  form 
yir  =  ax2r~l.  Accordingly,  all  curves  comprised  in  the  equation  aym  =  xmtl  are 
rectifiable,  m  being  any  integer.     (Compare  Art.  62). 

155.  The  Ellipse. — The  simplest  expression  for  the  arc 
of  an  ellipse  is  obtained  by  taking  x  =  a  sin  0,  whence 

y  =  b  cos  <j),  and  ds  =  {a2  cos20  +  b2  sin2#)*  d<p  ; 

.'.  s  =     (a2  cos2^  +  b2  sin20)*aty. 
It  is  often  more  convenient  to  write  this  in  the  form 

e2  sin20)*%  (3) 


•{C 


e  being  the  eccentricity  of  the  ellipse. 

It  may  be  observed  that  0  is  the  complement  of  the  eccen- 
tric angle  belonging  to  the  point  (#,  y). 

The  length  of  an  elliptic  quadrant  is  represented  by  the 
definite  integral 


a\    (1  -  e2  sin20)M0. 


We  postpone  the  further  consideration  of  elliptic  arcs  to 
a  subsequent  part  of  the  Chapter. 

156.  Rectification  in  Polar  Co-ordinates. — If  the 

curve  be  referred  to  polar  co-ordinates  we  plainly  have  (Diff. 
Calc,  Art.  180)  ds2  =  dr2  +  r2dd2 ;  hence  we  get 


8 


((-£)'".  *'-J('^    « 


Rectification  in  Polar  Co-ordinates.  227 

For  example,  the  length  of  the  spiral  of  Archimedes,  r  =  aO, 
is  given  by  the  equation 


=  -[(r*  +  a*)hdr. 


Comparing  this  with  the  formula  (2)  for  the  parabola,  it 
follows  that  the  length  of  any  arc  of  the  spiral,  measured 
from  its  pole,  is  equal  to  that  of  a  parabola  measured  from  its 
vertex. 

Examples. 

1.  Cardioid,  r  =  a(i  +  cos  0). 

dr 
Here  —  =  -  a  sin  0,  and  hence 

0  0 

s  =  a  J  { (1  4-  cos  0)2  +  Bm26}idd  =  2a  J cos  -  dd  =  ±a  sin  -  ■+  constant. 

2  2 

The  constant  becomes  zero  if  we  measure  *  from  the  point  for  which  0  =  o. 

2.  Logarithmic  spiral,  r  -  aO. 

Here,  if  b  =  ■ ,  we  get 

'  log  a         ■ 


r^0 


I ;     .-.  S  =  J"1  (i  +  $2)*rfr  =  (i  +  ft*)l(n  -  r0). 


Accordingly,  the  length  of  any  arc  is  proportional  to  the  difference  between 
the  vectors  of  its  extremities ;  a  result  which  also  follows  immediately  from  the 
property  that  the  curve  cuts  its  radius  vector  at  a  constant  angle. 


dr 

Taking  the  logarithmic  differentials,  we  get  — —  =  -  tan  mO  ; 

rad 

ds 
.•.  — - -  =  sec  md. 
rdd 

f  i-1 

Hence  «  =  «      (cosw0)m    dO. 

Or,  writing  <f>  for  mQi 

a  r  »« 

*  =  -j  (cos^)       d<p. 

This  is  readily  integrated  when  —  is  an  integer  (see  Art.  56). 

[15  a] 


228  Lengths  of  Curves. 

Whatever  be  the  value  of  m,  we  can  express  the  complete  length  of  a  loop  of 
the  curve  in  Gamma  Functions.  For  if  we  integrate  between  o  and  -,  we  ob- 
viously get  the  length  of  half  the  loop. 

Hence  the  length  of  the  loop  (Art.  122)  is 


s 


-r(-L) 


V    2m    J 


157.  Formula    of   Legend  re    on    Rectification. — 

Another  formula*  of  considerable  utility  in  rectification  fol- 
lows immediately  from  the  result  obtained  in  Art.  192,  Diff. 
Calo.    For,  if  this  result  be  written  in  the  form 

-^ — ■  =P,  we  get  s  -  t  =  fpdw.  (5) 

Consequently,  the  total  increment  of  s  -  t  between  any  two 
points  on  a  curve  is  equal  to  j  pda>  taken  between  the  same 
two  points. 

For  example,  in  the  parabola  we  have  p  =  ,  and 

r  cos  01 


hence 


t  =  a\  — —  =  a  log  tan  ( 
J  cos  &>  \ 


+  -  ]  +  const. 
4      2, 


If  we  measure  the  arc  from  the  vertex  of  the  curve,  and 

dt) 

observe  that  t  =  ~,  this  gives 


a  sin 


cos' 


w  ,  ,  fir  (i)\ 
-  +  a  log  tan  (-  +  -). 
m  \4      2) 


The  student  can  without  difficulty  identify  this  result  with 
that  given  in  Art.  151. 


*  This  theorem  is  due  to  Legendre.     See  Traite  dea  Fonctions  Elliptiques, 
tome  ii.,  p.  588. 


FagnanVs  Theorem. 


229 


It  should  be  observed  that  when  the  curve  is  closed,  its 
whole  length  is,  in  general,  represented  by 

r2ir 

pdw. 

Equation  (5)  furnishes  a  simple  method  of  expressing  the 
intrinsic  equation  of  a  curve,  when  we  are  given  its  equation 
in  terms  of  p  and  w. 

For,  if  p  =/(w)  we  have 


du) 


\pdio  =/((*))  +    /(<d)  da), 


(6) 


taken  between  suitable  limits. 

158.  Application  to  Ellipse.    Fagnani's  Theorem. 

In  the  ellipse  we  have 

p2  =  a2  cos2a>  +  b2  sin2w. 

Hence,  measuring  the  arc 
from  the  vertex  A,  and  observ- 
ing that  in  this  case  PiV'is  to  be 
taken  with  a  negative  sign,  we 
have 

arc  AP  +  PN  =  j    {a2  cos2a>  +  b2  sin2  w)l  dw, 

where  a  =  lACN. 

But,  in  Art.  155,  we  have  found  that  if  0  be  measured 
from  the  vertex  B,  the  arc  is  represented  by 


(a2  cos20  +  b2  sin20)*cfy. 

Consequently,  if  we  make  L  BCQ  =  a  =  L  AQN,  and  draw 
QM  perpendicular  to  the  axis  major  meeting  the  curve  inP', 
we  shall  have 

arc  BF  =  arc  AP  +  PN, 
or,  taking  away  the  common  arc  PPf, 

BP-APf  =  PN.  (7) 


230  Lengths  of  Curves.  t 

This  remarkable  result  is  known  as  Fagnanns  Theorem*, 
and  shows  that  we  oan  in  an  indefinite  number  of  ways  find 
two  arcs  of  an  ellipse  whose  difference  is  expressible  by  a  right 
line. 

We  add  a  few  properties  connecting  the  points  P  and  2* 
in  this  construction. 

Examples. 

i.  If  (x,  y)  and  {x\  y')  be  the  co-ordinates  of  P  and  P',  respectively ;  prove 
the  following : — 

(i).  «T«  — ,        (a).  PN  =  P'N',        (3).  CN.  CN'  =  CA  .  CB, 

(4).  op*  +  cn,o~  =  ca*  +  cb*=cp*+  cm. 

2.  Divide  an  elliptic  quadrant  into  two  parts  whose  difference  shall  be  equal 
to  the  difference  of  the  semiaxes. 

This  takes  place  when  P  and  P'  coincide  ;  in  which  case  CN  =  ^/ab,  and 
PN  =  a-b. 

We  shall  designate  the  point  so  determined  on  the  elliptic  quadrant  as  Fag- 
nani's  point. 

3.  Show  that  if  a  tangent  be  drawn  at  Fagnani's  point,  the  intercepts 
between  its  point  of  contact  and  its  points  of  intersection  with  tlio  axes  are 
respectively  equal  in  length  to  the  semi- axes  of  the  ellipse. 

4.  If  the  lines  PN  and  P'N'  be  produced  to  meet,  show  that  they  intersect 
on  the  confocal  hyperbola  which  passes  through  the  points  of  intersection  of  the 
tangents  to  the  ellipse  at  its  vertices.  Show  also  that  this  hyperbola  cuts  the 
ellipse  in  Fagnani's  point. 

*  Fagnani,  Giornale  de'  Letter ati  d' Italia,  17 16,  reprinted  in  his  Produzioni 
Matematiche,  1750.  It  may  be  noted  that  if  we  integrate  the  equation  of  Art. 
1 16,  Biff.  Calc,  taking  the  angle  C  as  obtuse,  and  adopting  zero  for  the  lowest 
limit  in  each  integral,  we  obtain 

J     \/i  -Wsv&ada  +  f    */ 1  -frBvxtbdb 

=  J    s/ 1  -k2Bm"cdo  +  k2  sin  a  sin*  sine, 

where  k  is  defined  by  the  equation  sin  C  =  k  sin  f,  and  a,  b,  c  are  connected  by 
the  relation 

cos  e  =  cos  a  cos  b  -  sin  a  sin  5yi  -  k*  sin2c. 

This  equation  furnishes  a  relation  between  three  elliptic  arcs,  from  which 
Fagnani's  theorem  can  be  readily  deduced,  as  well  as  many  other  theorems  con- 
nected with  such  arcs.     See  Legendre,  Pone.  Ellip. ,  tome  i.,  ch.  9. 


The  Hyperbola. 


231 


The  equation  of  PN  is 


x  sin 9  +  y  cos  d  =  \/a*  sin*0  +  b2  cos20, 


and  that  of  P'N'  is 


x  cos  0      y  sin  i 


If  we  eliminate  6,  we  get 


*      i/» 


which  represents  the  hyperbola  in  question. 

159.  The  Hyperbola. — In  the  hyperbola  we  have 

p2  =  a2  cos2  o)  -  b2  sin2  w. 

Hence,  measuring  the  arc  from  the  vertex  A  of  the  curve, 
we  find,  since  <o  is  measured  below  the  axis, 


PN-AP  = 


(«2cos2a>  -  62sin2a>)^a> 


where  a  =  LACN. 

As  we  proceed  along  the  hyperbola 
the  perpendicular  p  diminishes,  and 
vanishes  when  the  tangent  becomes 
the  asymptote. 

Moreover,  as  the  limit  of  w  in  this 

case  becomes  tan-1  T,  it  follows  that  the 
0 

diiference  between  the  asymptote  and 

the  infinite  hyperbolic  arc,  measured 

from  the  vertex,  is  represented  by  the 

definite  integral 

rtan-i^ 

{a2  cos2  ay  -  b2  sin2  <*>)  %dw . 


Fig.  29. 


Examples. 

1.  If  a  >  b,  prove  that 

/(«  +  b  cos  <p)id<p 

is  represented  by  an  elliptic  arc,  and  that  the  semiaxes  of  the  ellipse  are  the 
greatest  and  least  values  of  (a  +  b  cos  </>)*. 

2.  If  a<b,  prove  that 

I  {a  +  b  cos  <j>)ld<f> 

is  represented  by  the  difference  between  a  right  line  and  a  hyperbolic  arc. 


232  Lengths  of  Curves. 

1 60.  IJanden's  Theorem  on  a  Hyperbolic  Arc. — 

We  next  proceed  to  establish  an  important  theorem,  due  to 
Landen  ;*  namely,  that  any  arc  of  a  hyperbola  can  be  expressed 
in  terms  of  the  arcs  of  two  ellipses. 

This  can  be  easily  seen  as  follows : — In  any  triangle, 
adopting  the  usual  notation,  we  have 

c  =  aoosB  +  boos  A. 

Now,  representing  by  C  the  external  angle  at  the  vertex 
C,  we  have  C  =  A  +  B,  and  hence 

cdC '  =  (acosB  +  b  cos  A)  dA  +  (acosB  +  b  cos  A)  dB. 

Consequently,  supposing  the  sides  a  and  b  constant,  and 
the  remaining  parts  variable,  we  have 

\cdC  =    aco&BdA  +     b  cos  AdB  +  2asmB  +  const., 
or 

\ya2  +  b2  +  2abGO&CdC=  k/«2-^sin2^  dA+\^b*-a2sia2B  dB 

+  2a  sin  5  +  const.  (9) 

Now,  if  we  suppose  a  >  bA  */a2  -  b2sm2A  dA  represents 
(Art.  155)  the  arc  of  an  ellipse,  of  axis  major  2a  and  eccen- 
tricity -.  Also  \\/b2  -  a2sm2BdB  represents  (Art.  159)  the 
difference  between  a  right  line  and  the  arc  of  a  hyperbola, 
whose  axis  major  is  b  and  eccentricity  -,. 

/  Q  Q 

Again,  */a2  +  b2  +  2ab  eosC  =  J(a  -  b)2  sin2  -+(a  +  J)2  cos2  -, 


*  Landen,  Philosophical  Transactions,  1775 ;    also,  Mathematical  Memoirs, 
1780. 


Landerfs  Theorem  on  a  Hyperbolic  Arc.  233 

and  consequently  the  integral 


^/a2  +  b2  +  zab  cos  CdC 


represents  an  arc  of  the  ellipse  whose  semiaxes  are  a  +  b  and 
a  -b. 

Hence,  Landen's  theorem  follows  immediately. 

It  should  be  noted  that  the  limiting  values  of  A,  B  and 
O  are  connected  by  the  relations 

flsini?  =  J  sin  A,  and  C  =  A  +  B. 

Again,  if  we  suppose  the  angle  A  to  increase  from  o  to  7r, 
the  external  angle  G  will  increase  at  the  same  time  from 
o  to  7r,  while  B  will  commence  by  increasing  from  o  to  a, 

and  afterwards  diminish   from  a  to  of  where  a  =  sin-1-). 

Moreover,  in  the  latter  stage  b  cos  A  is  negative,  and  dB  also 
negative,  consequently  the  term  b  cos  A  dB  is  positive  through- 
out the  entire  integration  ;  and  the  total  value  of 

</b2  -  a?8>m2BdB  is  represented  by  2     ^/b2  -  a2  sin2 BdB. 

C 

Hence,  substituting  <p  for  — ,  and  integrating  between  the 

limits  indicated,  we  get,  after  dividing  by  2, 
it 

V {{a  +  b)2 sin2 <p  +  (a  -  b)2 cos2<t>}l df 


IT 

=  [V  -  V  &xl%A)UA  +  [V  -  a2  &m2B)l  dB.        (10) 

Accordingly,  the  difference  between  the  length  of  the  asymp- 
tote and  of  the  infinite  arc  of  a  hyperbola  is  equal  to  the  differ- 
ence between  two  elliptic  quadrants.  This  result  is  also  due  to 
Landen. 

We  next  proceed  to  two  important  theorems,  which  may 
be  regarded  as  extensions  of  Fagnani's  theorem. 


234 


Lengths  of  Curves. 


161.  Theorem*  of  Dr.  Graves. — If  from  any  point 
P  on  the  exterior  of  two  confocal  ellipses,  tangents  PT  and 
PTf  be  drawn  to  the  in- 
terior, then  the  difference 
(PT+PT'-TT')  between 
the  sum  of  the  tangents 
and  the  aro  between  their 
points  of  contact  is  con- 
stant. 

For,  draw  the  tangents 
QS  and  QS'  from  a  point 
Q,  regarded  as  infinitely 
near  to  P,  and  drop  the 
perpendiculars    PN     and  Fis-  3°- 

Qi\T ;  then,  since  the  conies  are  confocal,  we  have 

L  PQN=L  QPJST;  .-.  PN'  =  QN. 

Also,    PT=TR  +  RN=TR  +  RS+ SN-TS+SN 

=  T8+SQ-  QJST. 

In  like  manner 

PT'  =  PN'  +  S'Q-  T'S'; 

.-.  PT  +  PT  =  QS  +  QS'  +  TS-  T'S', 

or  PT  +  PT  -  TT  =  QS  +  QS'  -  SS'. 

Hence,  PT  +  PT'  -  TT'  does  not  change  in  passing  to 
the  consecutive  point  Q ;  which  proves  that  PT  +  PT'  -  TT' 
has  a  constant  value. 


*  This  elegant  theorem  was  arrived  at  hy  Dr.  Graves,  now  Bishop  of  Limerick, 
for  the  more  general  case  of  spherical  conies,  from  the  reciprocal  theorem,  viz. : — 
If  two  spherical  conies  have  the  same  cyclic  arcs,  then  any  arc  touching  the 
inner  will  cut  from  the  outer  a  segment  of  constant  area.  (See  Graves'  transla- 
tion of  Chasles  on  Cones  and  Spherical  Conies,  p.  77,  Dublin,  1841.) 

It  should  he  remarked  that  the  theorems  of  this  and  of  the  following  article 
were  investigated  independently  hy  M.  Chasles.  The  student  will  find  in  the 
Comptes  Rendus,  1843,  1844,  a  numher  of  beautiful  applications  by  that  great 
geometrician  of  these  theorems,  as  well  to  properties  of  confocal  conies,  as  also 
to  the  addition  of  elliptic  functions  of  the  first  species. 


Theorem  of  Dr.  Graves. 


235 


This  value  can  be  readily  expressed  by  taking  the  point 
at  B\  one  of  the  extremities 
of  the  minor  axis  of  the 
exterior  ellipse.  Let  D  be 
the  point  of  contact  of  the 
tangent  drawn  from  If,  and 
drop  Dif,  and  DN  perpen- 
dicular to  CA  and  CB, 
respectively. 

Let  CA  =  a,  CB  =  b, 
CA'=a',  CB'=b',  e  the  eccen- 
tricity of  interior  ellipse. 
Then,  by  Art.  155,  the  length  of  arc 


Fig-  3i" 


BD  =  a 


where 
Again, 


(1  -  e2sin20)Mc/), 


COS  a 


DM      CJST     CB      b 


hence 


CB      CB      CB'     r 

BD2  =  BN2  +  DN2  =(b'-b  cos  a)2  +  a2  sin2a 

■C'.-JiM-*). 


&D  m  j7</b*  -  P  -  <t  sin  a. 


Consequently  we  have 

B'D  -  BD  =  a'  sin  a  -  a\  (1  -  e2  sin2 
Hence,  in  general, 
PT+PT'-  TT  =  id  sin  a  -  2a  f"(i  -  e2  sin2tf>)^0, 

i> 


:«) 


where 


a  =  cos-1 


236 


Lengths  of  Curves. 


The  analogous  theorem,  due  to  Professor  Mac  Cullagh, 
may  be  stated  as  follows  : — 

162.  Theorem.— If  tangents  PT,  PT  be  drawn  to  an 
ellipse  from  any  point  on  a  con- 
focal  hyperbola,  then  the  differ- 
ence of  the  tangents  is  equal  to 
the  difference  of  the  arcs  TiTand 
KT. 

The  proof  is  left  to  the  student, 
and  is  nearly  identical  with  that 
given  for  the  previous  theorem. 

This  result  still  holds  when 
the  tangents  are  drawn  from  a 
point  on  an  ellipse  to  a  confocal 
hyperbola,  provided  that  the  tan- 
gents both  touch  the  same  branch 
of  the  hyperbola ;  as  can  be  seen 
without  difficulty. 

As  an  application*  we  shall  prove  another  theorem  of 
Landen;  viz.,  that  the  difference  between  the  length  of  the 
asymptote  and  of  the  infinite  branch  of 
a  hyperbola  can  be  expressed  in  terms 
of  an  arc  of  the  hyperbola. 

For,  let  the  tangent  at  A  meet 
the  asymptote  in  Z),  and  suppose  a 
confocal  ellipse  drawn  through  D. 
Then,  regarding  DT  as  a  tangent  to 
the  hyperbola,  it  follows,  by  the 
theorem  just  established,  that  the 
difference  between  DT  and  KT  is 
equal  to  the  difference  between  DA 
and  AK. 

Consequently  the  difference  be- 
tween the  asymptote  CT  and  the 
hyperbolic  branch  AT  is  equal  to 
DA  +  DC  -  zKA.  Consequently  the 
required  difference  is  expressible  in 


Fig.  33- 


terms  of  given  lines  and  of  the  hyperbolic  arc  AK, 


*  I  am  indebted  to  Dr.  Ingram  for  this  application  of  Professor  M'Cullagh's 
theorem. 


The  Epitrochoid  and  Hypotrochoid.  237 

"We  next  proceed  to  consider  two  important  curves  whose 
rectification  depends  on  that  of  the  ellipse. 

163.  The  Tiimat'oii. — From  the  equation  of  the  limacon, 

dv 
r  =  a  cos  0  +  ft,  we  get  -r=  =  -  a  sin  0, 

and  hence 

ds  =  (a2  +  62  +  2ab  cos  6)$dQ ; 

.-.  s  =  [  |(«  +  b)2  cos2  -  +  («  -  by  sin2 !j^0. 

Accordingly,  the  rectification  of  the  limacon  depends  on 
that  of  the  ellipse  whose  semiaxes  are  a  +  b  and  a  -  b. 

164.  The   Epitrochoid    and    Hypotrochoid. — The 

epitrochoid  is  represented  by  the  equations  (see  Diff.  Calc, 
Art.  284) 

x  =  (a  +  b)  cos  0  -  c  cos  — - —  0, 

J  r\     •     a  .      a  +  b  n 

y  =  {a  +  ft)  sin  0  -  c  sin  — 7—  0. 

Hence 

dx       ,        x  (  .    Q     c   .    a  +  b 


!=(«+6)jco80-^ 

0-4  &.) 

COS  —  0J. 

Squaring  and  adding  we  get 

(£)'-('-$(*» 

-  20c  cos  — 

ft 

■••-•-{-•Jh'-' 

.         00)  i 

20CCOS-t->    G 

tf0. 

ft) 

Hence,  substituting  — *  for  0,  we  get 
a 

2(a+b) 


s  = 


f  {(ft  +  c)2  sin20  +  (ft  -  c)2  cos20}i<fy. 


238  Lengths  of  Curves. 

Consequently  the  length  of  an  arc  of  the  epitrochoid  is  equal 
to  that  of  an  ellipse. 

The  corresponding  form  for  the  hypotrochoid  is  obtained 
by  changing  the  sign  of  b. 

165.  Steiner's  Theorem  on  Rectification  of 
Roulettes. — If  any  curve  roll  on  a  right  line,  the  length 
of  the  arc  of  the  roulette  described  by  any  point  is  equal 
to  that  of  the  corresponding  arc  of  the  pedal,  taken  with 
respect  to  the  generating  point  as  origin. 

For  (see  fig.  20,  Art.  145),  the  element  00' of  the  roulette 
is  equal  to  OPdio. 

Again,  to  find  the  element  of  the  pedal.  Since  the  angles 
at  N  and  N'  are  right,  the 
quadrilateral  NN'TO  is  inscri- 
bable  in  a  circle,  and  consequently 
NN'  =  OT  sin  NON'.  But,  in 
the  limit,  NN'  becomes  the  ele- 
ment of  the  pedal,  and  0Tbecomcs 
OP  :  hence  the  element  of  pedal 
is  OPdw ;  consequently  the  ele- 
ment of  the  pedal  is  equal  to  the 
corresponding     element     of    the  Fig.  34. 

roulette ;  .*.  &c. 

We  proceed  to  point  out  a  few  elementary  examples  of  this 
principle.  In  the  first  place  it  follows  that  the  length  of  an 
arc  of  the  cycloid  is  the  same  as  that  of  the  cardioid ;  and 
the  length  of  the  trochoid  as  that  of  the  limacon.  Again,  if 
an  ellipse  roll  on  a  right  line,  the  length  of  the  roulette 
described  by  either  focus  is  equal  to  the  corresponding  arc  of 
the  auxiliary  circle. 

Moreover,  it  is  easily  seen,  as  in  Art.  146,  that,  if  one 
curve  roll  on  another,  the  elements  ds  and  ds',  of  the  roulette, 
and  of  the  corresponding  pedal  are  connected  by  the  relation 


ds  =  ds- 


ft*  p\ 


In  the  case  of  one  circle  rolling  on  another,  this  relation 
shows  that  the  arcs  of  epicycloids  and  of  epitrochoids  are 
proportional  to  the  arcs  of  cardioids  and  of  limacons,  which 
agrees  with  the  results  established  already. 


Oval  of  Descartes. 


239 


1 66.  Oval  of  Descartes. — We  next  proceed  to  the 
rectification  of  the  Ovals  of  Descartes,  some  properties  of 
which  curves  we  have  given  in  chapter  xx.,  Diif.  Calc. 

The  curve  is  de- 
fined as  the  locus  of 
a  point  whose  dis- 
tances, rand  r',from 
two  fixed  points  are 
connected  by  the 
equation 

mr  +  1/  =  d, 

where  /,  m9  d  are 
constants. 

For  convenience 
we  shall  write  the 
equation  in  the  form 

mr  +  lr  =  nc,  (12) 

where  c  is  the  dis- 
tance between  the 
fixed  points.  g* 35' 

The  polar  equation  of  the  curve  is  easily  got.     For,  let  F 
and  Fl  be  the  fixed  points,  and  L  F\FP  =  0,  then  we  have 
/2  _  r2  +  c2  _  2rc  cog  Q  . 

also  from  (12), 

Pr'2  =  (nc  -  mr)2, 
hence  the  polar  equation  of  the  locus  is  readily  seen  to  be 

2  -  I2 
-„  =  o. 


mn  -  I2  cos  0       nn* 

2rc — - —  +  <r  — : 


mr 


mr 


(■3) 


For  simplicity  we  shall  write  this  in  the  form 

r2  -  2r£l  +  (7=  o.  (14) 

Solving  this  equation  for  r,  we  get 
r  =  Q  +  </&-  C,  otFF1  =  Q  +  >/QF^~C,  FP=Q-  </&  -  C. 

It  can  be  seen  without  difficulty  that,  so  long  as  /,  m,  n  are 
real  and  unequal,  the  curve  consists  of  two  ovals,  one  lying 
inside  the  other,  as  in  the  figure. 


240  Lengths  of  Curves. 

Again  we  get  from  (14),  by  differentiation 

(r  -  Q)  dr  =  rQ'dB,     where  O'  =  -^  ; 


.*.  — 77;  = —  =  — .  ;  nence  — -n  = —         — . 

rdd      r  -  Q      y&  _  Q  rdO         y^  _  q 


or       ds  =  qa/Q2  +  q/2    °dQ  ±  SQ*+Q*-Cde,  (15) 

V  G2  —  (7 

the  upper  sign  corresponding  to  the  outer  oval,  and  the  lower 
to  the  inner. 

Hence  the  difference  between  the  two  corresponding 
elementary  arcs  is  equal  to 

2v/iFTfl^J,     or,  2</a2  +  2ab  cos 6 +  b2-CdO, 

(writing  Q,  in  the  form  a  +  b  cos  0) ;  this  plainly  represents 
the  element  of  an  ellipse.  Consequently,  the  difference 
between  two  corresponding  arcs  of  the  ovals  can  be  repre- 
sented by  the  arc  of  an  ellipse.  This  remarkable  theorem  is 
due  to  Mr.  W.  Eoberts  (Liouville,  1847,  P-  x95)-  Some  years 
after  its  publication  it  was  shown  by  Professor  Genocchi 
(Tortolini,  1864,  p.  97),  that  the  arc*  of  a  Cartesian  is  ex- 
pressible in  terms  of  three  elliptic  arcs. 

In  order  to  establish  this  result  we  commence  by  proving 
one  or  two  elementary  properties  of  the  curve. 

Suppose  a  circle  described  through  F,  Flt  and  P ;  and  let 
PQ  be  the  normal  at  P  to  the  oval,  meeting  the  circle  in  Q, 
and  join  FQ  and  F,Q  ;  then  let  z  FPQ  =  a>,  and  F,PQ  =  w' ; 

and  since  m  —  +  /  -7-  =  o,  we  have  I  sin  </  -  m  sin  w ; 
ds        ds 

.-.  FQ  :  FXQ  =  l:m. 


*  For  the  proof  of  this  theorem  given  in  the  text  I  am  indebted  to  Mr. 
Panton. 


The  Cartesian  Oval.  241 

Also,  since  mr  +  Ir  =  nc  ;  and  (by  Ptolemy's  theorem) 

FP .  FtQ  +  FXP  .FQ  -  FFX .  PQ, 
we  have 

FQ  =  F\Q==PQ 

I         m         n 

Hence,  denoting  the  common  value  of  these  fractions  by 
u,  we  have 

FQ  =  luj    FXQ  =  mu,    PQ  =  nu. 

Again 


dr            a'                                 y&2  -  C 
tan  to  =  — jt;  =  — ;  ;     .*.  cos  o>  =  — — . 

Hence  the  first  term  in  the  expression  for  ds  in  (15)  is 
equal  to 

QdO  c      mn  -  P  cos  9  7A 

=  — - dd. 

cos  a>     m*  -  r       cos  o> 

Again,  let       L  FPFX  ■»  f$      lPFxC  =  0, 

and  we  have  the  two  following  relations  between  the  angles 
%  *,  4> : 

<f>  =  0  +  $,     I  sin  0  +  m  sin  0  =  n  sin  i/>.  (16) 

Hence 

d(j>-  dQ  =  dip,     I  cos  Odd +  m  cos QdQ  =  n  cos  \pd\p  ; 
.-.  (mn  -  I2  cos  0)d0  =  m(n  +  lcos<l>)d<l>  -  n  (m  +  I  cos  ^)^, 


or 


#m  -  P  cos  0  _n         w  +  /  cos  0  ,         m  +  /  cos  \L  _ ,     .     , 

— 1 dv  =  m dd>-  n r  aw.    (17) 

COS  o>  COS  (O  cos  w 

Again,  from  the  triangle  FPQ,  we  have 

r  cos  (o  =  PQ  +  jPQ  cos  0  =  (n  +  I  cos  <f)u ; 

w  +  J  cos  0     r        y- s - - 

.\  -  =  -  =  v/J2  +  ri*  +  2/ft  cos  0. 

COSw  M  r 

[16] 


242  Lengths  of  Curves. 

In  the  same  manner  it  can  be  shown  that 

m+Zcosi//      c       /- ■ 7- 

=  -  =  v  P  +  rn 2  +  zlm  cos  \p. 


COS  (t) 

Hence  we  have 


[€ldQ  mc     f     j- = - . 

=  —5 — -    vl  +  n2  +  zln  cos  0  dd> 

Jcosw     m2  -  I2)  r    r 

tic      f     j  — 
- — -2    */l2  +  m*+2lm  cos  \pd\p.  ( 1 8) 

Each  of  these  latter  integrals  is  represented  by  the  arc  of  an 
ellipse,  and,  accordingly,  the  arc  of  a  Cartesian  Oval  is 
expressible  in  the  required  manner. 

It  should  be  noted  that  the  limiting  values  of  0,  <j>,  and  \p 
are  connected  by  the  relations  given  in  (16). 

Again,  it  can  be  shown  without  difficulty  that  the  axes  of 
the  ellipses  are  the  lines  (AB,'CD),  (AC,  BD),  and  (AD,  BC), 
respectively :  a  result  also  given  by  Signor  Gtenocchi.  First, 
with  respect  to  the  ellipse  whose  element  is  a/q?  +  Qf2  -  CdO, 
it  is  plain  that  its  axes  are  the  greatest  and  least  values  of 
2  ya2  +  Q,'2-C,  or  of  2V/a2  +  62+  iab  cos  0  -  C ;  but 


are  2*/ (a  +  h)2  -  C  and  2  </(a  -  b)2  -  C,  which  are  plainly 
the  same  as  the  greatest  and  least  values  of  Pl\  ;  and,  con- 
sequently, are  AB  and  CD. 

Again,  from  the  equation  mr  +  1/  =  nc,  we  get 

mFB  +  l(FB  +  e)  -  nc;      .-.  FB  =  &zB*. 

v  '  l  +  m 


In  like  manner, 


FC=(n  +  l)e. 
l+m 


Again,  since  we  get  the  points  on  the  outer  oval  by 
changing  the  sign  of  /,  we  have 

FAJ»±!)o     FDJnzIh 
m-l  m-l 


Rectification  of  Curves  of  Double  Curvature.         243 
and,  consequently, 


AD=™      BO     2m 


m-V  I  +  mf 

2mc(n  +  l)      7>T^_2mc(n-l)  t 
■A.0  = 5 — ™    y     JjJJ  —        o     72     J 

m2  -P  m2-  r 

but  these  are  readily  seen  to  be  the  values  for  the  axes  of  the 
ellipses  in(i8). 

It  should  be  noted  that  if  we  substitute  in  (15)  the  values 
for  a  and  b,  the  expression  for  the  element  ds  becomes  of  the 
following  symmetrical  form : 


mc       y- — = ; T .       nc 


*/ V + n2+ 2  In  cos  (j>d<p — 5 — "-//2+^+2/mcosif4 


?y  r  r    m2-l 

Ic 


T9  \/m2  +  n2  -  2tnn  cos  BdO.  (19) 

m2  -  I2 

We  shall  conclude  the  Chapter  with  a  brief  account  of 
the  rectification  of  curves  of  double  curvature. 

167.  Rectification  of  Curves  of  Double  Curvature. 

If  the  points  in  a  curve  be  not  situated  in  the  same  plane,  the 
curve  is  said  to  be  one  of  double  curvature.  The  expression 
for  its  length  is  obtained  in  an  analogous  manner  to  that 
adopted  for  plane  curves ;  for,  if  we  refer  the  curve  to  a 
system  of  rectangular  axes  in  space,  and  denote  the  co-ordi- 
nates of  two  consecutive  points  by  (#,  y,  2),  [x+dx,  y+dy,  z  +  dz), 
we  get  for  the  element  of  length,  ds,  the  value 

ds  =  */dx2  +  dy2  +  dz2. 

The  curve  is  commonly  supposed  to  be  determined  by  the 
intersection  of  two  cylindrical  surfaces,  whose  equations  are 
of  the  form 

f(x,y)  =0,        0(0,  s)  =0. 

From  these  equations,  if  -f  and  —  be  determined,  the  formula 
dx        dx 

of  rectification  is 

-JH(2HS)'f*       « 

[16  a] 


244  Lengths  of  Curves. 

When  2  is  taken  as  the  independent  variable,  this  formula 
becomes 


Ji'*®1*®)'- 


the  limits  being  in  each  oase  determined  by  the  conditions  of 
the  question. 

The  simplest  example  is  that  of  the  helix,  or  the  curve 
formed  by  the  thread  of  a  screw.  From  its  mode  of  generation 
it  is  easily  seen  that  the  helix  is  represented  by  two  equations 
of  the  form 


z\  .      z 


x  =  a  cos  \  7  l,      y  =  a  sin  .  . 


dx       a   .    fz\      dy     a 


Hence 

.\  ds=l  i  +  —  j  dz,    ors  =  (i  +  TjiJz; 

the  arc  being  measured  from  the  point  in  which  the  helix 
meets  the  plane  of  xy. 

This  result  can  also  be  readily  established  geometrically. 

Examples, 
i.  Find  the  length  of  the  curve  whose  equations  are 


a£  x* 

2(1 


'"-•     •-»• 


*-  »j('+5*£)M(i+S)*— »— •■ 

the  arc  being  measured  from  the  origin.  \ 

This  is  a  case  of  a  system  of  curves  which  are  readily  rectified ;  for,  in  ge- 
neral, whenever 

(dy\~  _     dz_ 

\dx)    ~2dz' 

I        dy*      dz*\h       /         dz\ 
we  have  [1+d*  +  d7>)    =  {l+Tx)> 

and  therefore  ds  =  dx  +  dz,    or    s  =  x  +  z  +  const. 


Rectification  of  Curves  of  Double  Curvature.  245 


Thus,  if  y  =f{x)  be  one  of  the  equations  of  a  curve,  we  get  —  =/'(*),  and 


hence,  if  a  second  equation  he  determined  from  the  equation 


dz       i 
dx      2 


/»)', 


the  length  of  the  curve  is  represented  by  x  +  z  +  const.  ;  the  value  of  the  con- 
stant being  determined  by  the  conditions  of  the  problem. 
For  instance,  if  y  —  a  sin  a;,  we  get  f'{x)  —  a  cos  a;,  and 

dz      a2  a? 

— -  =  —  cos2 a; ;    .*.  a  =  —  (x  +  cos x  sin  x). 

dx      i  4  x  ' 

Hence  the  length  of  the  curve  of  intersection  of  the  cylindrical  surfaces 


y  =  a  sin  x,        z  =  —  (x  +  cos  x  sin  x) 
4 

is  z  +  it)  the  length  being  measured  from  the  origin. 

/—                         2     & 
2.    y=2yax-x,    z  =  x J— .  -4n*.  *  =  *  +  y  -  z. 

o    ™    »* 


2  Z 


3.     —  -  ^  =  I,     a;  =  -  («°  +  e  °),  the  length  being  measured  from  the  point 


of  intersection  of  the  curve  with  the  plane  of  xy. 


(a2  +  b2)l 


246  Lengths  of  Curves. 

Examples. 

i.  Find  the  length  of  any  arc  of  the  catenary 

a  I  ■       _5\ 
</  =  -[**  +  *  »), 

and  show  that  the  area  between  the  curve,  the  axis  of  x,  and  the  ordinates  at 
two  points  on  the  curve,  is  equal  to  a  times  the  length  of  the  arc  terminated  by 
those  points. 

f       v  df 
a.  In  any  curve  prove  that  *  =    ——==,  and  hence  find  the  length  of  a 

parabolic  arc. 

3.  Show  that  the  integral  l  may  be  represented  by  an  arc  of 

J  \/bx*  -&-& 
a  circle,  and  find  the  limiting  values  of  x  for  its  possibility. 

Wa2  —  e1  x2 
— 2 — r  *x* 

where  a  is  the  semiaxis  major,  and  e  the  eccentricity. 

5.  Express  the  length  of  an  elliptic  quadrant  in  a  series  of  ascending  powers 
of  its  eccentricity. 

6.  Prove  that  the  integral  of 

x2dx 


yV-j32)(a2-s») 

can  be  represented  by  an  arc  of  the  ellipse  whose  semiaxes  are  o  and  £. 

7.  Show  that  the  rectification  of  the  sinusoid  y  =  b  sin  x  is  the  same  as  that 
of  an  ellipse. 

8.  Prove  that  the  whole  length  of  theirs*  negative  pedal  of  an  ellipse,  taken 
with  respect  to  a  focus,  is  equal  to  the  circumference  of  the  circle  described  on 
the  axis  minor  as  diameter. 

9.  Show  that  the  length  of  an  arc  of  the  curve  r  =  a  sin  nd  is  equal  to  that 
of  an  arc  of  the  ellipse  whose  semiaxes  are  a  and  na. 

10.  If,  from  the  equation  of  a  curve  referred  to  rectangular  co-ordinates,  we 
form  an  equation  in  polar  co-ordinates,  by  taking  r  =  y  and  rdd  =  dx,  then  the 
lengths  of  the  corresponding  arcs  of  the  two  curves  are  equal,  and  the  area  J  y  dx 
of  the  former  curve  is  equal  to  the  corresponding  sectorial  area  of  the  latter. 

11.  Prove  that  the  difference  between  the  lengths  of  the  two  loops  of  the 
limacon  r  =  a  cos  d  +  b  is  equal  to  8b :  a  being  greater  than  b. 

12.  Being  given  three  points  A,  B,  C  on  the  circumference  of  an  ellipse, 
show  that  we  can  always  find,  at  either  side  of  C,  a  fourth  point  B  such  that  the 
difference  between  AB  and  CD  shall  be  equal  to  a  right  line. 


Examples.  247 

13.  If  a  circle  be  described  touching  two  tangents  to  an  ellipse  and  also 
touching  the  ellipse,  prove  that  the  point  of  contact  with  the  ellipse  divides  the 
elliptic  arc  between  the  points  of  contact  of  the  tangents  into  two  parts,  whose 
difference  is  equal  to  the  difference  of  the  lengths  of  the  tangents  (Chasles, 
Comptes  Bendus,  1843). 

14.  Prove  that  the  entire  length  of  any  closed  curve  is  represented  by 

I  - —  taken  round  the  entire  curve  ;  p  being  the  radius  of  curvature  at  any 
point,  and  p  the  length  of  the  perpendicular  from  any  fixed  point  on  the  tangent. 


ex  + 


02x 


15.  If  ev  =  be  the  equation  of  a  curve,  prove  that  — -  =  - ,  and 

e*  —  1  dz     e2x  -  1 

hence  rectify  the  curve. 

16.  Calculate  approximately,  by  the  tables  of  Art.  125,  the  whole  length  of 

a  loop  of  the  curve  r  =  a   cos  -  6. 

Here,  by  Ex.  3,  Art.  156,  the  required  length  is 

,  /-r(f)      yd) 

2rtA/ '     —>—(-,  or  2«a/ it  — ^ — L  . 

*      r(?)  r(') 

Hence,  taking  logarithms,  and  observing  that  —  =  1.625,  and  -=  1.125,  we 

o  o 

get  as  the  required  approximation  a  x  3.29488.     The  figure  of  this  curve  is 
exhibited  in  Art.  268,  Diff.  Calc. 

17.  In  a  Cartesian  Oval  whose  two  internal  foci  coincide,  prove  that  the 
difference  of  the  two  arcs,  intercepted  by  any  two  transversals  from  the  exter- 
nal focus,  is  equal  to  a  straight  line  which  may  be  found.  [The  above  curve 
is  the  inverse  of  an  ellipse  from  a  focus.] — Professor  Crofton,  Educ.  Times, 
June,  1874. 

From  (13)  Art.  166,  it  follows,  making  n  =  w,  that  the  equation  of  the 
limaqon,  in  this  case,  is 

I2  cos  9  -  m2 
P  -  m2 
which  is  of  the  form 

r2  +  2r(a  cos  $  -  $)  +  (a  -  j8)2  =  o. 

Hence,  by  (15),  the  difference  between  two  corresponding  elementary  arcs  is 

/—      e 

4V  aj8  cos  -  dO. 

Consequently,  if  0i  and  02  be  the  values  of  0  for  the  two  transversals  in 
question,  we  get  the  difference  of  the  corresponding  arcs 


=  8^  oj8(  sin sin-  1 . 


Also,  it  can  be  readily  seen  that  the  distance  between  the  vertices  of  the 
limacon  is  4\/  aj8  ;  .-.  &c. 


248  Lengths  of  Curves. 

xz     v2 
1 8.  Show  that  the  length  of  an  arc  of  the  ellipse  -z  +  \-  -  i  is  represented 

a1      b2 
by  the  integral 

"•J 


(a2cos20  +  £2sin20)* 
we  have  ds  =  pdd,  an 
9.  Show,  in  like  manner,  that  the  length  of  a  hyperbolic  arc  is  represented 


a2b2 
This  result  is  easily  seen,  for  we  have  ds  =  pdd,  and  p  =  — -  ;  .-.  &c 


by 

*<•( dl 

J  (a8cos20-i2sin20)* 
20.  Hence  prove  that  the  integral 

dx 


\~- 


bx2)*(a!  -b'x2f 


is  represented  by  an  elliptic  arc  when  ab'  >  ba\  and  by  a  hyperbolic  arc  when 
ab'  <  ba'.    . 

21.  Prove  that  the  differential  of  the  arc  of  the  curve  found  by  cutting  in 
the  ratio  n  :  1  the  normals  to  the  cycloid 

y  =  a  +  b  cos  u,    x  =  au  +  b  sin  u, 


^j(a  +  nb) 


2  +  4nab  sin2  -  du. 
2 


22.  Each  element  of  the  periphery  of  an  ellipse  is  divided  by  the  diameter 
parallel  to  it :  find  the  sum  of  all  the  elementary  quotients  extended  to  the  entire 
ellipse.  Ans.  it. 

23.  In  the  figure  of  Art.  158,  if  a  =  L  ACN',  and  £  =  L  BCN,  prove  that 

tan  a  _  tan  £ 

24.  Find  the  length,  measured  from  the  origin,  of  the  curve 

v 
x*  =  a2(i  -  ea). 

Ans.  s  =  a  log  ( J  -  x. 

0  \a-x) 

25.  Find  the  length,  measured  from  <p  =0,  of  the  curve  which  is  represented 
by  the  equations 

x  —  (2a  —  b)  sin  <p  -  (a  —  b)  sin3^>, 

y  =  (2b  —  a)  cos  <p  -  (b  -  a)  cos3</>. 

Ans.  s  =  \(a  +  b)<p  +  f  (a  -  b)  sin  <p  cos  <p. 

26.  Prove  that  the  sides  of  a  polygon  of  maximum  perimeter  inscribed  in  a 
conic  are  tangents  to  a  confocal  conic. — Chasles,  Comptes  Jftendus,  1845. 

27.  To  two  arcs  of  an  equilateral  hyperbola,  whose  difference  is  rectifiable, 
correspond  equal  arcs  of  the  lemniscate  which  is  the  pedal  of  the  hyperbola. 
Ibid. 


Examples.  249 

28.  The  tangents  at  the  extremities  of  two  arcs  of  a  conic,  whose  difference 
is  rectifiable,  form  a  quadrilateral,  whose  sides  are  tangents  to  the  same  circle. — 
Ibid. 

29.  In  an  equilateral  hyperbola  prove  that 

rds  =  %a2d  (tan  26), 

and  hence  show  that  $rds  taken  between  any  two  points  on  the  curve  is  equal  to 
the  rectangle  under  the  chord  joining  the  points  and  the  line  connecting  the 
middle  point  of  the  chord  with  the  centre  of  the  hyperbola.     Mr.  W.  S.  M'Cay. 

30.  If 

z  +  s3  z  —  z3 

x  =  a  —    ,,  y  =  a -. 

1  +  a*'  *         I  +  z4 

be  any  point  on  a  curve,  show  that  the  arc  is  the  integral  of 

r,    /Z  — =  (M.  Serret 

What  curve  do  the  equations  represent  ? 

31.  Through  any  point  in  a  plane  two  conies  of  a  confocal  system  can  be 
drawn.  If  the  distance  between  the  foci  be  2c,  and  the  transverse  semi-axes  of 
these  conies  be  fx,  v,  prove  the  following  expression  for  any  arc  of  a  curve 

ds2  =  (jji?  -  1/-)  I  — + 

Kn  t/x2  -  c2      c2  -  v 

32.  Prove  that  the  following  relation  is  satisfied  by  the  \x  and  v  of  any  point 
on  a  tangent  to  the  ellipse  for  which  ft  has  the  value  /*i : 

dp  dv 


VV  -  *)  J?  -  /t!2)         y/{?  -  V2)  (Ml2  -  V2) 

33.  The  arc  of  the  envelope  of  the  right  line  x  sin  a  -  y  cos  o  =/(a)  is  the 
integral  of  (/(o)  +  /"  (o))  da.  (Hermite,  Cours  d' Analyse.) 

34.  The  arc  of  the  curve  in  which  y2  +  a2  x2  -  zax  -  o  and  zs  -  b2  x2  +  2bx  =  o 
intersect,  if  a2  =  1  +  b2,  is 

V  2  (a  -  b)dx 


«y  x  (2  —  ax)  (2  -  bx) 


{Ibid). 


xm      yin 
35.  Show  that  the  arc  of  the  curve  —  +  —  =  1  depends  on  an  integral  of 


the  form 


f  dz  \Za*  (1  +  z)k  +  **(!  -  z)*,   where  h  =  — 


36.  Show  that  rectification  may,  in  general,  be  reduced  to  quadratures  as 
follows : — 

Produce  each  ordinate  of  the  curve  to  be  rectified  until  the  whole  length  is  in 
a  constant  ratio  to  the  corresponding  normal  divided  by  the  old  ordinate,  then 
the  locus  of  the  extremity  of  the  ordinate  so  produced  is  a  curve  whose  area  is  in 
a  constant  ratio  to  the  length  of  the  given  curve. 

By  this  theorem  Van  Huraet  rectified  the  semi-cubical  parabola  nearly  simul- 
taneously with  Wm.  Neil. 


(    250    ) 


CHAPTER  IX. 


VOLUMES    AND    SURFACES    OF    SOLIDS. 


1 68.  Solids. — The  Prism  and  Cylinder. — The  most 
simple  solid  is  the  cube,  which  is  accordingly  the  measure  of 
all  solids,  as  the  square  is  that  of  all  areas.  Hence  the 
finding  the  volume  of  a  solid  is  called  its  cubature.  Before 
proceeding  to  the  application  of  the  Integral  Calculus  to 
finding  the  volumes  and  surfaces  of  solids  we  propose  to  show 
how,  in  certain  cases,  such  volumes  and  surfaces  can  be  found 
from  geometrical  considerations.  In  the  first  place,  the 
volume  of  a  rectangular  parallelepiped  is  measured  by  the 
continued  product  of  the  three  adjacent  edges ;  and  that  of 
any  parallelepiped  by  the  area  of  a  face  multiplied  by  its 
distance  from  the  opposite  face. 

Again,  the  volume  of  a  right  prism  is  measured  by  the 
product  of  its  altitude  into  the  area  of  its 
base.  For  example,  the  volume  of  the  right 
prism  represented  in  the  figure  is  mea- 
sured by  the  area  of  the  polygon  ABODE, 
multiplied  by  the  altitude  AA!.  Again, 
since  each  lateral  face,  AB  B'A'  for  ex- 
ample, is  a  rectangle,  it  follows  that  the 
sum  of  the  areas  of  all  the  faces  (exclusive 
of  the  two  bases),  i.e.  the  area  of  the  sur- 
face of  the  prism,  is  equal  to  the  rectangle 
under  the  altitude  and  the  perimeter  of 
the  polygon  which  forms  its  base. 

This  and  the  preceding  result  still  hold 
in  the  limit,  when  the  base,  instead  of  a  polygon,  is  a  closed 
curve  of  any  form,  in  which  case  the  surface  generated  is 
called  a  cylinder.  Hence,  if  V  denote  the  volume  of  the  por- 
tion of  a  cylinder  bounded  by  two  planes  drawn  perpendi- 
cular to  its  edges,  h  its  height,  and  A  the  area  of  its  base,  we 
get  V=Ah. 


rig.  36. 


The  Pyramid  and  Cone.  251 

Again,  if  2  denote  the  superficial  area  of  a  cylinder, 
bounded  as  before,  and  S  the  length  of  the  curve  which  forms 
its  base,  we  have  S  =  Sh. 

169.  The  Pyramid  and  Cone. — If  the  angular  points 
of  a  polygon  be  joined  to  any  external  point,  the  solid  so 
formed  is  called  &  pyramid.  Any  section  of  a  pyramid  by  a 
plane  parallel  to  its  base  is  a  polygon  similar  to  that 
which  forms  the  base,  and  the  ratio  of  their  homologous 
sides  is  the  same  as  that  of  the  distances  of  the  planes  from 
the  vertex  of  the  pyramid.  Hence  it  follows  that  pyramids 
standing  on  the  same  base,  and  whose  vertices  lie  in  a  plane 
parallel  to  the  base,  are  equal  in  volume.  For,  the  sections 
made  by  any  plane  parallel  to  the  base  are  equal  in  every 
respect  ;  and,  consequently,  if  we  suppose  the  pyramids 
divided  into  an  indefinite  number  of  slices  by  planes  parallel 
to  the  base,  the  volumes  of  the  corresponding  slices  will  be 
the  same  for  all  the  pyramids ;  and  hence  the  entire  volumes 
are  equal. 

Also,  if  two  pyramids  have  equal  altitudes,  but  stand  on 
different  polygonal  bases,  the  volumes  of  the  pyramids  will 
be  to  each  other  in  the  same  proportion  as  the  areas  of  the 
polygonal  bases.  For,  this  proportion  holds  between  the 
areas  of  the  sections  made  by  any  plane  parallel  to  the  base ; 
and  consequently  between  the  slices  made  by  two  infinitely 
near  planes. 

Again,  the  pyramid  whose  base  is  one  of  the  faces  of  a 

cube,   and  whose  vertex  is  at  the  centre   of  the   cube,  is 

the  one-sixth  part  of  the  cube ;  for  the  entire  cube  can  be 

divided  into  six  equal  pyramids,  one  for  each  face.     Hence, 

denoting  the  side  of  a  cube  by  a,  the  volume  of  the  pyramid 

#3 
in  question  is  represented  by  — ;  i.  e.  by  the  product  of  the 

area  of  its  base  into  one-third  of  its  height. 

Now,  if  we  vary  the  base,  without  altering  the  height, 
from  what  has  been  established  above  it  follows  that  the 
volume  of  any  pyramid  is  the  area  of  its  base  multiplied  by 
one-third  of  its  height* 


*  This  demonstration  is  taken  from  Clairaut's  Elimens  de  Geometric  The 
student  is  supposed  familiar  with  the  more  ancient  proof,  from  the  property  that 
a  triangular  prism  can  be  divided  into  three  pyramids  of  equal  volume. 


252  Volumes  and  Surfaces  of  Solids. 

If  the  base  of  the  pyramid  be  any  closed  curve,  the  solid 
so  formed  is  called  a  cone  ;  and  we  infer  that  the  volume  of  a 
cone  is  equal  to  one-third  of  the  product  of  the  area  of  its  base 
into  its  height. 

If  the  base  of  a  pyramid  be  a  regular  polygon,  and  the 
vertex  be  equidistant  from  the  angular  points  of  the  polygon, 
the  pyramid  is  called  a  right  pyramid. 

In  this  case  each  face  of  the  pyramid  is  an  isosceles  triangle, 
whose  area  is  the  rectangle  under  the  side  of  the  polygon 
and  half  the  perpendicular  of  the  triangle.  Hence  the 
surface  of  the  pyramid  is  equal  to  the  rectangle  under  the 
semi-perimeter  of  the  regular  polygon  and  the  perpendicular 
common  to  each  face  of  the  pyramid. 

Again,  if  we  suppose  the  number  of  sides  of  the  regular 
polygon  to  become  infinite,  the  pyramid  becomes  a  right 
cone ;  and  we  infer  that  the  entire  surface  of  a  right  cone  is 
equal  to  the  rectangle  under  the  semi-circumference  of  its 
circular  base  and  the  length  of  an  edge  of  the  cone. 

Hence,  if  a  be  the  semi-angle  of  the  cone,  I  the  length  of 
an  edge,  and  r  the  radius  of  its  base,  wo  have  r  =  I  sin  a,  and 
the  surface  of  the  cone  is  represented  by  it  I2  sin  a. 

If  a  right  cone  be  divided  by  two  planes  ABC,  DEF, 
perpendicular  to  its  axis,  as  in  figure,  the  0 

part  intercepted  by  the  planes  is  called  a 
truncated  cone. 

The  surface  of  a  truncated  cone  is 
easily  expressed ;  for  if  OA  =  /,  OD  =  I', 
the  required  surface  is  -n  sin  a  (I2  -  I'2), 


ac  >c 


L' 

or  it  {I-  l')(l+  0  sin  a. 

Now,  if  the  circular  section  LMN  be      L /--  -\n 

drawn  bisecting  the  distance   between      /  m 

ABC&n.di  DEF,  the  circumference  of  the 
circle  LMN  is  iz  (I  +  V)  sin  o.  Hence  the 
surface  of  the  truncated  cone  is  equal  to  b 

the  rectangle  under  the  edge  AD  and  the  Fig.  37. 

circumference  of  LMN  its  mean  section. 

1 70.  Surface  and  Volume  of  a  Sphere. — To  find  the 
superficial  area  of  a  sphere ;  suppose  a  regular  polygon  in- 
scribed in  a  semicircle,  and  let  the  figure  revolve  around  the 
diameter  AB  ;  then  each  side  of  the  polygon,  PQ  for 
example,  will  describe  a  truncated  cone. 


Surface  and  Volume  of  a  Sphere. 


253 


Fig.  38. 


Now,  from  the  centre  C  draw  CD  perpendicular  to  PQ, 
and  construct,  as  in  figure  ;  then,  by  the  preceding  Article, 
the  surface  generated  hjPQ  is 
equal  to  2ir  PQ  .  DI. 

Again,  by  similar  triangles, 
we  have  DC:DI=PQ:  MN; 
.-.  PQ.DI=DC.MN. 

Accordingly,  since  the  per- 
pendicular CD  is  of  same  length 
for  each  side  of  the  polygon,  the 
surface  generated  by  the  entire 
polygon  in  a  complete  revo- 

lution  is  equal  to  2  tt  CD  .  AB  =  47r  R2  cos  -  ;  where  n  repre- 

n 

sents  the  number  of  sides  of  the  polygon,  and  R  the  radius  of 

the  circle. 

If  we  suppose  n  to  become  infinite,  the  solid  generated 
by  the  polygon  becomes  a  sphere  ;  and  we  get  qirli*  for  the 
entire  surface  of  the  sphere.  Hence,  the  surface  of  a  sphere 
is  equal  to  four  times  the  area  of  one  of  its  great  circles. 

Again,  it  is  easy  to  find  the  surface  generated  by  any 
number  of  sides  of  the  polygon.  Thus,  for  example,  that 
generated  by  all  the  sides  lying  between  the  points  A  and  Q 
is  plainly  equal  to  2ir  CD  .  AN". 

Hence,  in  the  limit,  the  surface  generated  in  a  complete 
revolution  by  the  arc  AQ  is  equal  to  2tt  .  AC  .  AN.  Such  a 
portion  of  a  sphere  is  called  a  spherical  cap. 

Again,  suppose  the  points  A  and  Q  connected  ;  then,  since 
A  Q3  =  A B  .  AN,  it  follows  that  the  area  of  the  spherical  cap 
generated  by  the  arc  AQ  is  equal  to  the  area  of  the  circle 
whose  radius  is  the  chord  AQ. 

The  volume  of  a  sphere  is  readily  found  from  its  surface ; 
for  we  may  regard  the  volume  as  consisting  of  an  infinitely 
great  number  of  pyramids,  having  their  common  vertex  at 
the  centre,  and  whose  bases  form  the  entire  surface.  But  the 
volume  of  each  pyramid  is  represented  by  the  product  of  one- 
third  of  its  height  (i.  e.  the  radius)  by  its  base.  Hence  the 
entire  volume  of  the  sphere  is  one-third  of  its  radius  multi- 
plied by  its  surface,  i.  e.  —  R3, 


25  i  Volumes  and  Surfaces  of  Solids. 


Examples. 

1.  If  a  sphere  and  its  circumscribing  cylinder  be  cut  by  planes  perpendi- 
cular to  the  axis  of  the  cylinder,  prove  that  the  intercepted  portions  of  the 
surfaces  are  equal  in  area. 

2.  Prove  that  the  volume  of  a  sphere  is  to  that  of  its  circumscribing  cylinder 
in  the  proportion  of  2  to  3  :  and  that  their  surfaces  also  are  in  the  same  propor- 
tion.    These  results  were  discovered  by  Archimedes. 

171.  Surfaces  of  Revolution. — In  the  preceding  we 
have  regarded  a  sphere  as  generated  by  the  revolution  of  a 
circle  around  a  diameter.  In  general,  if  any  plane  be  sup- 
posed to  revolve  around  a  fixed  line  situated  in  it,  every  point 
in  the  plane  will  describe  a  circle,  and  any  curve  lying  in  the 
plane  will  generate  a  surface. 

Such  a  surface  is  called  a  surface  of  revolution ;  and  the 
fixed  line,  round  which  the  revolution  takes  place,  is  called 
the  axis  of  revolution. 

It  is  obvious  that  the  section  of  a  surface  of  revolution 
made  by  any  plane  drawn  perpendicular  to  its  axis  is  a 
circle. 

If  we  suppose  any  solid  of  revolution  to  be  cut  by  a  series 
of  planes  perpendicular  to  its  axis,  the  volume  of  the  solid 
intercepted  between  any  two  such  sections  may  be  regarded 
as  the  limit  of  the  sum  of  an  indefinite  number  of  thin  cylin- 
drical plates. 

Now,  if  we  suppose  the  generating  curve  to  be  referred  to 
rectangular  axes,  the  axis  of  revolution  being  that  of  x,  the 
area  of  the  circle  generated  by  a  point  (x,  y)  is  plainly  equal 
to  iry2,  and  the  cylindrical  plate  standing  on  it,  whose  thick- 
ness is  dx,  is  represented  by  iry^dx. 

Hence,  the  element  of  volume  of  the  surface  of  revolution 
is  ny2  dx,  and  the  entire  volume  comprised  between  two  sec- 
tions, corresponding  to  the  abscissae  a  and  j3,  is  obviously 
represented  by  the  definite  integral 

fdx, 

a 

in  which  the  value  of  y  in  terms  of  x  is  to  be  got  from  the 
equation  of  the  generating  curve. 


The  Sphere.  255 

In  like  manner,  the  volume  of  the  surface  generated  by 
the  revolution  of  a  curve  around  the  axis  of  y  is  represented 
by  ir\x2dy,  taken  between  suitable  limits. 

Again,  we  may  regard  the  surface  generated  by  any 
element  ds  of  the  curve  as  being  ultimately  a  portion  of  the 
surface  of  a  truncated  cone,  as  in  Art.  170;  and  hence  the 
surface  generated  by  ds  in  a  complete  revolution  round  the 
axis  of  x  is  represented  by  iiryds  ;  and  accordingly  the  entire 
surface  generated  is  represented  by 


27T 


[yds 


re- 


taken between  proper  limits. 

We  proceed  to  apply  these  f ormulse  to  a  few  elementary 
examples. 

172.  The  Sphere. — Let  x2  +  y2  =  ft2  be  the  equation  of 
the  generating  circle  ;  then,  substituting  a2  -  x2  for  y2s  we  get 
for  the  volume 

(a2  -  x2)  dx  =  7r  ( a2x ]  +  const. 

If  we  take  o  and  a  as  limits,  we  get for  the  volume  of 

3 

the  hemisphere  ;  .*.  the  entire  volume  of  the  sphere  is , 

o 
as  in  Art.  1 70. 

To  find  the  volume  of  a  spherical  cap,  let  h  be  the  length 
of  the  portion  of  the  diameter  cut  off  by  the  bounding  plane, 
and  we  get  for  the  corresponding  volume 

7r         (ft2  -  x2)  dx  =  nh2  ( ft  - 
}a-h  \        3, 

Again,  to  find  the  superficial  area,  we  have 


i+gY*  (- 

dx 


=  (  1  +  —  )dx  =  -  dx:  .*.  yds  =  adx. 


Hence,  the  surface  of  the  zone  contained  between  two 
parallel  planes  corresponding  to  the  abscissas  xx  and  xQ  is 


27r      adx  =  27rft  fa  -  x0) ; 

Jx0 


256  Volumes  and  Surfaces  of  Solids. 

that  is  the  product  of  the  circumference  of  a  great  circle  by 
the  breadth  of  the  zone.     This  agrees  with  Art.  1 70. 

173.  Right  Cone. — If  a  denote,  as  before,  the  angle 
which  the  right  line  which  generates  a  cone  makes  with  its 
axis  of  revolution,  we  get  y  =  x  tan  a,  taking  the  vertex  of  the 
cone  as  origin,  and  the  axis  of  revolution  as  that  of  x ;  accord- 
ingly, the  element  of  volume  is  ir  t&n2ax2dx. 

Hence,  if  h  denote  the  height  of  the  cone,  we  get  its 
volume  equal  to 

7r  tan2a      x2dx  =  —  tan2a ; 
Jo  3 

i.e.  -  x  area  of  its  base,  as  in  Art.  169. 
3 
Again,  to  find  its  surface,  we  have  ds  =  sec  adx ; 

r» 

.\  27r  j  yds  =  27r  tan  a  sec  a      xdx  =  irk7,  tan  a  sec  a  ; 

J  0 

which  agrees  with  the  result  already  obtained. 


Examples. 

1.  The  base  of  a  cylinder  is  a  circle  whose  area  is  equal  to  the  surface  of  a 
sphere  of  radius  5  ft. ;  being  given  that  the  volume  of  the  cylinder  is  equal  to 
the  sum  of  the  volumes  of  two  spheres  of  radii  9  ft.  and  16  ft.,  find  the  height 
of  the  cylinder.  Ans.  64J  ft. 

2.  A  solid  sector  is  cut  out  of  a  sphere  of  10  ft.  radius,  by  a  cone  the  angle 
of  which  is  1 200 ;  find  the  radius  of  the  sphere  whose  solid  contents  are  equal  to 
those  of  the  sector.  Ans.  sv  2. 

3.  Two  eones  have  a  common  base,  the  radius  of  which  is  12  ft.  ;  the  alti- 
tude of  one  is  9  ft.  ;  and  that  of  the  other  is  5  ft. ;  find  the  radius  of  a  sphere 
whose  entire  surface  is  equal  to  the  sum  of  the  areas  of  the  cones. 

Ans.  2*y zi  ft. 

174.  Paraboloid  of  Revolution. — Writing  the  equa- 
tion of  a  parabola  in  the  form  y2  =  zmx,  we  get  for  the 
volume  of  the  solid  generated  by  its  revolution  round  the 
axis  of  x 

2vm  f  xdx  =  nmx2  +  const.  =  -  y2x  +  const. 


Surface  of  Spheroid.  257 

Hence,  the  volume  of  the  surface  generated  by  the  revo- 
lution of  the  part  of  a  parabola  between  its  vertex  and  the 

point  (a?i,  yi)  is  represented  by  -  y2xlt  i.e.  is  equal  to  half  the 

volume  of  the  circumscribing  cylinder. 

Again,  to  find  the  surface  of  the  paraboloid,  we  have 

yds  =  yi  i  +  —  J  dy  =  -  (y2  +  m2)hydy. 

Hence,  the  surface  of  the  paraboloid,  between  the  same 
limits  as  above,  is  represented  by 


^  j  !  (y2  +  m*)*ydy  =  ^  j^2  +  mj- 


w 


175.  Spheroids  of  Revolution. — If  we  suppose  an 
ellipse  to  revolve  round  its  axis  major,  the  surface  generated 
by  the  revolving  curve  is  called  a  prolate  spheroid.  If  it  re- 
volve round  the  axis  minor  the  surface  is  called  an  oblate 
spheroid. 

The  volume  of  a  spheroid  is  easily  obtained  ;  for,  taking 

-z  +  j£  =  i  as  the  equation  of  the  curve,  we  get,  on  substitut- 
ing b2  ( 1  -  X-  J  for  y\ 


a2 


x2)dx  =  —r-x\a2 }  +  const, 


*•     \        3 


Hence  the  entire  volume  is  —  ab2.     In  like  manner,  the  vo- 

3 

lume  of  an  oblate  spheroid  is  obviously  —  ba2. 

176.  Surface  of  Spheroid. — In  the  case  of  a  prolate 
spheroid  we  have 

A         (  P*^ 7 

ds  =    1  +  — —    dx ; 
[17] 


258 


Volumes  and  Surfaces  of  Solids. 


Hence,  if  CN  =  xXi  CM =  x0,  we  get  for  S,  the  zone  gene- 
rated in  a  complete  revo- 
lution by  the  arc  PQ, 


S 


x2 )  dx. 


Now,  if  we  take  CD  =  - 

and   construct   an   ellipse 
whose   semiaxes    are   CD  Fig.  39. 

and  CD,  it  is  easily  seen 

(Art.  129)  that  the  elementary  area  between  two  consecutive 

be  fa2 


ordinates  of  this  ellipse  is 


a  \e 


■j 


x2    dx.      Hence  it  follows 


that  the  area  of  the  zone  generated  by  the  arc  PQ  is  tt  times 
the  area  of  the  portion  P1Q1Q2P2  of  this  ellipse. 

Again,  if  AEX  be  the  tangent  at  the  vertex  of  the  original 
ellipse,  we  see  that  the  entire  surface  of  the  spheroid  is  4ir 
x  the  area  BCAEX ;  but  this  is  seen,  without  difficulty,  to  be 


\7rb'  +  27r  —  sm  le. 
e 


(») 


In  like  manner,  we  get  for  the  surface  $  generated  by  the 
revolution  of  an  ellipse  round  its  minor  axis 


xds  =  27r    (  a 2  +  -Tj-  y2  J  dy 
a2e[ 


S=27T 


If  this  be  integrated,  as  in  Art.  151,  we  get,  after  some 
obvious  reductions, 

s  =  *f  («*y  +  *•)*  +  Jl  V  +  (aY + ffg. 

If  this  be  taken  between  the  limits  o  and  £>,  and  doubled,  we 
get  fo_:  the  entire  surface  of  the  ellipsoid 


27T«    +  7T 


^m 


CO 


Ellipsoid  of  Revolution.  259 

It  is  readily  seen,  as  in  the  former  case,  that  the  surface 
of  any  zone  of  this  ellipsoid  is  ic  times  the  area  of  a  corre- 
sponding portion  of  the  hyperbola 

x%     a2e2y2  _ 
a2  "  ~~br  "  l 

bounded  by  lines  drawn  parallel  to  the  axis  of  x. 

The  area  of  the  surface  generated  by  the  revolution  of  a 
hyperbola  round  either  axis  admits  of  a  similar  investigation. 


Examples. 

i.  Find  the  volume  of  tie  surface  generated  by  the  revolution  of  a  cycloid 
round  its  base.  r 

Here,  referring  the  cycloid  to  DA  and       WJL        _1B 
JDB  as  co-ordinate  axes,  we  have  (see  DifL 
Calc,  Art.  272) 


x  =  a(<p  +  sin^)),     y  =  a(i  +  cos<p) 

Hence 

d  V=  vy1  dx  =  it  a3  ( 1  +  cos  <pfd<p ;  Flg'  4°* 

.-.  for  the  entire  volume  V,  we  get 

J7T  c  If  <b 

(1  +  cos  $fd<p  =  i6ira3       cos^  -  dtp 
0  Jo2 

IT 

m  32ira3      cos6  0^0,    making  -  =  d. 
Jo  2 

Hence  F-5^«3- 

2.  Find  the  whole  surface  generated  in  the  same  case. 

Here  S  =  2ir  \y  ds  =  4«-a2  I  (1  4-  cos  <f>)  cos  -  d<p ; 

hence  the  entire  surface  is 

Jtf       0<b  .        64irfl2 
cos3  -d<j>  = . 

0     r      2    -1  5 

[17  a] 


260 


Volumes  and  Surfaces  of  Solids. 


3.  Find  the  volume  and  the  surface  of  the  solid  generated  by  the  revolution 
of  the  tractrix  round  its  axis. 

(1).  Here  we  have 

y*dx  =  -{di-y*)lydy; 

hence  the  volume  generated  by 
the  portion  AP  is 

*[* (a*-y*)lydy  =  -  (a*-y>)l. 
h  3 

The  volume  generated  by  the 

entire  tractrix  is  —  a3:  i.  e.  half 

3 

the  volume  of  the  sphere  whose  Fig.  41. 

radius  is  OA. 

(2).  The  surface  generated  by  AP  is 

2ir  I  yds  =  2ifl      dy     (see  Ex.  2,  Art.  134) 

=  2ira{a  -y). 

Hence  the  entire  surface  generated  is  2ira2 ;  i.  e.  half  the  surface  of  the  sphere 
of  radius  OA. 

4.  Find  the  volume,  and  also  the  surface,  generated  by  the  revolution  of  the 
catenary  around  the  axis  of  x. 

(1).  Here  the  volume  of  the  solid  gene-  ^ 
rated  by  VP  is  represented  by 

x\    y*dx=  —  (    (ea  +  e    •  +  z\dx 


ira  . 
=  —  (ys  +  ax), 


where  *  =  PV. 

(2).  Again,  since 

we  have 


Fig.  42. 


2t(  yda=  —  \  y%dx. 


Surfaces  of  Revolution. 


261 


Consequently  the  surface  generated  by  TV  in  a  complete   revolution  is  - 

x  the  volume  generated ;  i.e.  =  ir  (y«  +  ax). 

5.  In  the  same  curve  to  find  the  surface  generated  by  its  revolution  round 
the  axis  0  V. 
Here 


Again 


S  =  27r  I  xds  =  7r  i  xe°  dx  +  x  1  xe"adx. 


Also  the  value  of 


J*      •  5         r*  • 

xeadx  =  axea-  a  \    eadx=a\ 
0  Jo 

3  of 

f«     _* 

I    xe  adz 

Jo 


a;  1 


is  obtained  by  changing  the  sign  of  a  in  the  last  result. 
Hence 


xe  adx  =  a2  —  axe    "  —  a2e    a; 


I! 

•\  S  =  v  \ia2  +  ax  (e«-  e~"\  -  a2(e"  +  e~«\  J 

=  2ir(az  +  xs  —  ay). 

177.    Annular    Solids. — If   a  y 

closed  curve,  which  is  symmetrical 
with  respect  to  a  right  line,  be  made 
to  revolve  round  a  parallel  line,  then 
the  superficial  area  generated  in  a 
complete  revolution  is  equal  to  the 
product  of  the  length  of  the  moving 
curve  into  the  circumference  of  the 
circle  whose  radius  is  the  distance 
between  the  parallel  lines. 

This  is  easily  proved:  for  let 
APBP'  be  any  curve,  symmetrical  with  respect  to  AB,  and 
suppose  OX  to  be  the  axis  of  revolution ;  and  draw  FN,  QM 
two  indefinitely  near  lines  perpendicular  to  the  axis.  It  is  evi- 
dent that  PQ  =  P'Q'.  Again,  let  PJV=  y,  P'JV= tf,  PQ  =  P'Q' 
=  ds,  DN  =  b  ;  then  the  sum  of  the  elementary  zones  described 
by  PQ  and  P'Q'  in  a  complete  revolution  is  represented  by 

27T  (y  +  y')  ds  =  4irb  ds. 


Fig.  43- 


262  Volumes  and  Surfaces  of  Solids. 

Consequently  the  surface  generated  by  the  entire  curve  is 
2irbS,  where  S  denotes  the  whole  length  of  the  curve. 

A  similar  theorem  holds  for  the  volume  of  the  solid  ge- 
nerated :  viz.,  the  volume  generated  is  equal  to  the  product 
of  the  area  of  the  revolving  curve  into  the  circumference  of 
the  same  circle  as  before. 

For  the  volume  of  this  solid  is  plainly  represented  by 


jV-y,2)tf*, 


or  by.  7T    (y-y')(y  +  y')dx  =  2-nb\  (y-y')dz. 

But  the  area  of  the  curve  is  represented  by 

(y  -  y)  & : 


i 


consequently,  denoting  this  area  by  A,  and  the  volume  by  V, 
we  have 

V  =  zirb  x  A. 

In  these  results  the  axis  of  revolution  is  supposed  not 
to  intersect  the  curve ;  if  it  does,  the  expression  2nb  x  A 
represents  the  difference  between  the  volumes  of  the  surfaces 
generated  by  the  portions  of  the  curve  lying  at  opposite  sides 
of  the  axis  of  revolution  ;  as  is  readily  seen.  A  similar  alte- 
ration must  be  made  in  the  former  theorem  in  this  case. 

If  a  circle  revolve  round  any  external  axis  situated  in  its 
plane,  the  surface  generated  is  called  a  spherical  ring.  From 
the  preceding  it  follows  that  the  entire  surface  of  such  a  ring 
is  a^ab ;  where  a  is  the  radius  of  the  circle,  and  b  the  dis- 
tance of  its  centre  from  the  axis  of  revolution. 

In  like  manner  the  volume  of  the  ring  is  2ir2arb. 

It  would  be  easy  to  add  other  applications  of  these 
theorems. 

178.  ©uldln's*  Theorems. — The  results  established  in 
the  preceding  Article  are  but  particular  cases  of  two  general 


*  Guldin,  Centrobaryica,  seu  de  centro  gmvitatis  trium  specierum  quantitatis 
contimuc,  1635.  Guldin  arrived  at  his  principle  by  induction  from  a  small  num- 
ber of  elementary  cases,  but  his  attempt  at  a  general  demonstration  was  an 
eminent  failure.  See  Montucla  Hist,  des  Math.,  torn.  ii.  p.  34.  Montucla  has 
shown,  torn.  ii.  p.  92,  that  Guldin's  theorems  can  be  established  from  geome- 
trical considerations,  without  recourse  to  the  Calculus. 


Guldin's  Theorems.  263 

propositions,  usually  called  Gruldin's  Theorems,  but  originally 
enunciated  by  Pappus  (see  Walton's  Mechanical  Problems, 
p.  42,  third  Edition).     They  may  be  stated  as  follows  : — 

(1).  If  a  plane  curve  revolve  round  any  external  axis, 
situated  in  its  plane,  the  area  of  the  surface  generated  is  equal 
to  the  product  of  the  perimeter  of  the  revolving  curve  by  the 
length  of  the  path  described,  during  the  revolution,  by  the  centre 
of  gravity  of  that  perimeter . 

(2).  Under  the  same  circumstances,  the  volume  of  the  solid 
generated  is  equal  to  the  product  of  the  area  of  the  generating 
curve  into  the  path  described  by  the  centre  of  gravity  of  the  re- 
volving area. 

To  prove  the  former,  let  s  denote  the  whole  length  of  the 
curve,  x,  y,  the  co-ordinates  of  one  of  its  points,  x,  y,  those 
of  the  centre  of  gravity  of  the  curve ;  then,  from  the  defi- 
nition of  these  latter,  we  have 

.'.  2-nrys  =  27r  j  yds, 

i.  e.  the  surface  generated  by  revolution  round  the  axis  of  x  is 
equal  to  the  product  of  8,  the  length  of  the  generating  curve, 
into  2-n-y,  the  path  described  by  the  centre  of  gravity. 

To  prove  the  second  proposition ;  let  A  denote  the  area 
of  the  generating  curve,  and  dA  the  element  of  area  corre- 
sponding to  any  point  x,  y.  Also  let  x,  y  be  the  co-ordinates 
of  the  centre  of  gravity  of  the  area,  then 

y  =  — ^j—  =         A       (substituting  dx  dy  for  dA) ; 

.;.  2-n-yA  =  2-rr  jjydxdy  =  irj  y2dx; 

where  the  integral  is  supposed  taken  for  every  point  round  the 
perimeter  of  the  curve  :  but,  from  Art.  171,  the  integral  at 
the  right-hand  side  represents  the  volume  of  the  solid  gene- 
rated ;  hence  the  proposition  in  question  follows. 

For  example,  tho  volume  of  the  ring  generated  by  the 
revolution  of  an  ellipse  around  any  exterior  line  situated  in 
its  plane  is  at  once  2iriabc,  where  a  and  b  are  the  semiaxe3 


264  Volumes  and  Stir/aces  of  Solids. 

of  the  ellipse,  and  c  is  the  distance  of  its  centre  from  the  axis 
of  revolution. 

It  may  be  noted  that  these  results  still  hold  if  we  suppose 
the  curve,  instead  of  making  a  complete  revolution,  to  turn 
round  the  axis  through  any  angle.  For,  let  0  be  the  circular 
measure  of  the  angle  of  rotation,  and  in  the  former  case  we 
have 

Oys  =  0  j  yds. 

But  By  is  the  length  of  the  path  described  by  the  centre 
of  gravity,  and  0  j  yds  is  the  area  of  the  surface  generated  by 
the  curve ;  .*.  &o. 

In  like  manner  the  second  proposition  can  be  shown  to 
hold. 

Again,  (xuldin's  theorems  are  still  true  if  we  suppose  the 
rotation  to  take  place  around  a  number  of  different  axes  in 
succession ;  in  which  case  the  centre  of  gravity,  instead  of 
describing  a  single  circle,  would  describe  a  number  of  arcs  of 
circles  consecutively ;  and  the  whole  area  of  the  surface  ge- 
nerated will  still  be  measured  by  the  product  of  the  length  of 
the  generating  curve  into  the  path  of  its  centre  of  gravity  ; 
for  this  result  holds  for  the  part  of  the  surface  corresponding 
to  each  axis  of  revolution  separately,  and  therefore  holds  for 
the  sum. 

Again,  in  the  limit,  when  we  suppose  each  separate  rota- 
tion indefinitely  small,  we  deduce  the  following  theorem.  If 
any  plane  curve  move  so  that  the  path  of  its  centre  of  gravity 
is  at  each  instant  perpendicular  to  the  moving  plane,  then  the 
surface  generated  by  the  curve  is  equal  to  the  length  of  the 
curve  into  the  path  described  by  its  centre  of  gravity. 

The  corresponding  theorem  holds  for  the  volume  of  the 
surface  generated. 

These  extensions  of  Guldin's  theorems  were  given  by 
Leibnitz  {Act.  Erud.  Lips.,  1695). 

179.  Expression  for  Volume  of  any  Solid. — The 
method  given  in  Art.  1 7 1  of  investigating  the  volume  bounded 
by  a  surface  of  revolution  can  be  readily  extended  to  a  solid 
bounded  in  any  manner.  For,  if  we  suppose  the  volume 
divided  into  slices  by  a  system  of  parallel  planes,  the  entire 
volume  may,  as  before,  be  regarded  as  the  limit  of  the  sum 


Volume  of  Elliptic  Paraboloid.  265 

of  a  number  of  infinitely  thin  cylindrical  plates.  Thus,  if  we 
suppose  a  system  of  rectangular  co-ordinate  axes  taken,  and 
the  cutting  planes  drawn  parallel  to  that  of  xy  ;  then,  if  Az 
represent  the  area  of  the  section  made  by  a  plane  drawn  at 
the  distance  z  from  the  origin,  the  entire  volume  is  denoted 

by 

f  Azdz, 

taken  between  proper  limits. 

The  area  Az  is  to  be  determined  in  each  case  as  a  function 
of  z  from  the  conditions  of  the  bounding  surface. 

For  example,  to  find  the  volume  of  the  portion  of  a  cone 
cut  off  by  any  plane ;  we  take  the  origin  at  the  vertex,  and 
the  axis  of  z  perpendicular  to  the  cutting  plane  ;  then,  if  B 
denote  the  area  of  the  base,  and  h  the  height  of  the  cone,  it 
is  easily  seen  that  we  have 

Bz2 
Az:  B=z2  :  h2,  or  As  =  -—  ; 
hr 

B[h  i 

.*.   V  =  —\    z2dz  =  -  B  x  h:  as  in  Art.  160. 

If  the  cutting  planes  be  parallel  to  that  of  yz,  the  volume 
is  denoted  by  f  Axdx;  where  Ax  denotes  the  area  of  the  sec- 
tion at  the  distance  x  from  the  origin. 

1 80.  Volume   of  Elliptic   Paraboloid.— Let    it  be 

proposed  to  find  the  volume  of  the  portion  of  the  elliptic 
paraboloid 

x2     y2 

-  +-  =  2Z, 

p     q 

cut  off  by  a  plane  drawn  perpendicular  to  the  axis  of  the  sur- 
face.    Here,  considering  z  as  constant,  the  area  of  the  ellipse 

—  +  —  =  22,  by  Art.  128,  is  iirz^/pq. 

Hence,  denoting  by  c  the  distance  of  the  bounding  plane 
from  the  vertex  of  the  surface,  we  have 


V  =  2ir*/pq     zdz  =  ire2  */pq. 

J  n 


266  Volumes  and  Surfaces  of  Solids. 

This  result  admits  of  being  exhibited  in  another  form  ;  for  if 
B  be  the  area  of  the  elliptic  section  made  by  the  bounding 
plane,  we  have 

B  =  2nc^/pq. 

Hence  V  =  J  circumscribing  cylinder,  as  in  paraboloid  of  re- 
volution. 

181.  The  ellipsoid.— Next,  to  find  the  volume  of  the 
ellipsoid 

x2     y2     z2 


The  section  of  the  surface  at  the  distance  z  from  the  origin 
is  the  ellipse 

or      if  z2 

—  +  —  =  i : 

a2     b2  c2 

the  area  of  this  ellipse  is 

ab,  i.e.  Az  =  7r(  i  ~  —  )ab. 

Hence,  denoting  the  entire  volume  by  V,  we  have 

f*  /       s2\         4 
V=27rab\    (  i -  —  )dz  =-Trabc. 

JoV       cV  3 

182.  Case  of  Oblique  Axes. — It  is  sometimes  more 
convenient  to  refer  the  surface  to  a  system  of  oblique  axes. 
In  this  case,  if,  as  before,  we  take  the  cutting  planes  parallel 
to  that  of  27/,  and  if  w  be  the  angle  the  axis  of  2  makes  with 
the  plane  of  xy,  the  expression  for  the  volume  becomes 

sin  to  j  Azdz, 

taken  between  proper  limits,  where  Az  represents  the  area  of 
the  section,  as  in  the  former  case. 

For  example,  let  us  seek  the  volume  of  the  portion  of  an 
ellipsoid  cut  off  by  any  plane. 


7T     I 


Case  of  Oblique  Axes.  267 

Suppose  DED'E'  to  represent  the  section  made  by  the 
plane,  and  ABA'B'  the  parallel  central  section.  Take  OA, 
OB,  the  axes  of  this  section  as  axes  of 
x  and  y  respectively  ;  and  the  conju- 
gate diameter  OC  as  axis  of  z. 

Then  the  equation  of  the  surface 


is 


x2       y2      z2  _ 


where  OA  =  a',  OB  =  b\  00  -  c'. 

It  will  now  be  convenient  to  transfer  the  origin  to  the 
point  C,  without  altering  the  directions  of  the  axes,  when  the 
equation  of  the  surface  becomes 


x2      y      2%       2 
aT2  +  Vi  =  V~"c7 


The  area  Az  of  the  section,  by  Art.  128,  is 

™'K?-S);  (3) 

hence,  denoting  C'N  by  h,  the  volume  cut  off  by  the  plane 
BED'  is  represented  by 


ira 
or 


'6'sinwf„4(7-^)'/s' 


ira'b'  sin  to 

But,  by  a  well-known  theorem,*  we  have 

db'cr  sin  w  =  «fo, 

where  a,  b,  c,  are  the  principal  semiaxes  of  the  surface. 

Hence  the  expression  for  the  volume  V  in  question  be- 
comes 

F--:«^5-£);  (4) 

*  Salmon's  Geometry  of  Three  Dimensions,  Art.  96. 


268  Volumes  and  Surfaces  of  Solids, 


C'N 
or,  denoting  ^  by  k, 


V  =  irabck2  f  I 


-;} 


This  result  shows  that  the  volume  cut  off  is  constant  for  all 

sections   for  which  k  has  the  same  value.      Again,   since 

OiV 

■jrp,  =  i  -  k,  the  locus  of  iVis  a  similar  ellipsoid ;  and  we  infer 

that  if  a  plane  cut  a  constant  volume  from  an  ellipsoid,  the  locus 
of  the  centre  of  the  section  is  a  similar  and  similarly  situated 
ellipsoid. 

183.  Elliptic  Paraboloid. — The  corresponding  results 
for  the  elliptic  paraboloid  can  be  deduced  from  the  preceding 
by  adopting  the  usual  method  of  such  derivation :  viz.,  by 
taking 

a}  =pc,     b2  =  qc, 

and  afterwards  making  c  infinite ;  observing  that  in  this  case 
the  ratio  -  becomes  unity. 

Making  these  substitutions  in  (4),  it  becomes 

V  =  7r  y^pqh2  (  1 A  or  irh2  */pq,  since  d  =  00. 

Hence,  if  a  constant  length  be  measured  on  any  diameter 
of  an  elliptic  paraboloid  and  a  conjugate  plane  drawn,  then 
the  volume*  of  the  segment  cut  from  the  paraboloid  by  the 
plane  is  constant. 

Again,  the  area  of  an  elliptic  section  by  (3)  is 

,T,f2h     h2\         nabc   (ih     h2\ 

^h{v-72)0X7^Z>Kl'Y2) 


*  For  a  more  direct  investigation  the  student  is  referred  to  a  memoir  "  On 
some  Properties  of  the  Paraboloid,"  Quarterly  Journal  of  Mathematics,  June, 
1874,  by  Professor  Allman. 


Elliptic  Paraboloid.  269 

On  making  the  same  substitutions,  this  becomes  for  the 
paraboloid 

27r  */pq 


h. 


sin  u) 


Now,  if  we  suppose  a  cylinder  to  stand  on  this  section, 
the  volume  of  the  portion  cut  off  by  the  parallel  tangent 
plane  to  the  paraboloid  is  obtained  by  multiplying  the  area 
of  the  section  by  h  sin  w ;  and,  consequently,  is 


27T 


*/pqh\ 


i.  e.  is  double  the  corresponding  volume  of  the  paraboloid. 
This  is  an  extension  of  the  theorem  of  Art.  1 80. 

Examples. 

1.  Prove  that  the  volume  of  the  segment  cut  from  a  paraboloid  by  any  plane 
is  f  ths  of  that  of  the  circumscribing  cone  standing  on  the  section  made  by  the 
plane  as  base. 

2.  A  cylinder  intersects  the  plane  of  xy  in  an  ellipse  of  semiaxes  OA  =  a, 
OB  =  b,  and  the  plane  of  xz  in  an  ellipse  of  semiaxes  OA  =  a,  00  =  c ;  the 
edges  of  the  cylinder  being  parallel  to  BO ;  find  the  volume  of  the  portion  of  the 
cylinder  bounded  by  the  three  co-ordinate  planes.  Am.  \  abc. 

3.  The  axes  of  two  equal  right  cylinders  intersect  at  right  angles ;  find  the 
volume  common  to  both.  Ans.  A3&  a3,  where  a  is  the  radius  of  either  cylinder. 
This  surface  is  called  a  Groin. 

184.  'Volume  by  Double  Integration. — In  the  ap- 
plication of  the  preceding  method  of  finding  volumes  the 
area  represented  by  Ax,  instead  of  being  immediately  known, 
requires  in  general  a  previous  integration ;  so  that  the  deter- 
mination of  the  volume  of  a  surface  involves  two  successive 
integrations,  and  consequently  V  is  expressed  by  a  double 
integral. 

Thus,  as  the  area  Ax  lies  in  a  plane  parallel  to  that  of  yzy 
its  value,  as  in  Art.  126,  may  generally  be  represented  by 
\  zdy,  taken  between  proper  limits.  Hence  F"may  be  repre- 
sented by 

or,  adopting  the  usual  notation,  by 
Jjzdydx9 
taken  between  limits  determined  by  the  data  of  the  question. 


270 


Volumes  and  Surfaces  of  Solids. 


The  value  of  z  is  supposed  given  by  a  relation  2  =/(#,  y), 
by  means  of  the  equation  of  the  bounding  surface ;  hence 

fzdy-lf(x,y)dy. 

In  the  determination  of  this  integral  we  regard  x  as 
constant  (since  all  the  points  in  the  area  have  the  same 
value  of  #),  and  integrate  with  respect  to  y  between  its  proper 
limits. 

Thus,  if  yx  and  y0  denote  the  limiting  values  of  y,  the 
definite  integral 

becomes  a  function  of  x  :  this  function,  when  integrated 
with  respect  to  x  between  the  proper  limits,  determines  the 
volume  in  question. 

If  X\  and  x0  denote  the  limits  of  x,  V  may  be  represented 
by  the  double  integral 


dydx. 


We  shall  exemplify  this  by  a  figure,  in  which  we  suppose 
the  volume  bounded  by  the  plane  of  xy,  by  a  cylinder 
perpendicular  to  that  plane,  and 
also  by  any  surface.*  Let 
RPJXQ  represent  the  section  of 
the  cylinder  by  the  plane  of  xy ; 
and  suppose  PMNQ  to  be  the 
section  of  the  volume  by  a  plane 
parallel  to  yz  at  the  distance  x 
from  the  origin.  Let  PL  =  yu 
QL  =  y0,  then  the  area  PMNQ 
is  represented  by  the  integral 


*  The  determination  of  a  volume  of  any  form  is  virtually  contained  in  this. 
For,  if  we  suppose  the  surface  circumscribed  by  a  cylinder  perpendicular  to  the 
plane  of  xyy  the  required  volume  will  become  the  difference  between  two 
sylinders,  bounded  by  the  upper  and  lower  portions  of  the  surface,  respectively. 
See  Bertrand,  Calc.  Int.  §  447. 


Volume  by  Double  Integration.  271 

The  values*  of  yx  and  yQ  in  terms  of  x  are  obtained  from 
the  equation  of  the  curve  RPRtQ. 

Again,  suppose  P'M'N'Q'  to  represent  the  parallel  section 
at  the  infinitesimal  distance  dx  from  PMNQ,  then  the 
elementary  volume  between  PMNQ  and  P'M'N'Q!  is  repre- 
sented by 


dx 


zdy. 


Now,  if  R  T  and  R!Tr  be  tangents  to  the  bounding  curve, 
drawn  perpendicular  to  the  axis  of  x,  and  if  OT  =  xlf  OT=x0i 
the  entire  volume  is  represented  by 


z  dy  dx. 

J  *o  J  Vo 


It  should  be  observed  that  zdydx  represents  the  volume 
of  the  parallelepiped  whose  height  is  z,  and  whose  base  is  the 
infinitesimal  rectangle  having  dx  and  dy  as  sides  ;  and  conse- 
quently the  volume  may  be  regarded  as  the  sum  of  all  such 
parallelepipeds  corresponding  to  every  point  within  the  area 
RPR'Q. 

It  is  also  plain  that  we  shall  arrive  at  the  same  result 
whether  we  integrate  first  with  respect  to  x,  and  afterwards 
with  respect  to  y,  or  vice-versa ;  i.  e.  whether  we  conceive  the 
volume  divided  into  slices  parallel  to  the  plane  of  xz,  or  to 
that  of  yz. 

"We  shall  illustrate  the  preceding  by  an  example. f 

Suppose  RPR'Q  to  be  the  circle 

(x-ay+(y-by  =  R>, 

and  the  bounding  surface  the  hyperbolic  paraboloid 
xy  =  cz ; 


*  In  our  investigation  we  have  assumed  that  the  parallels  intersect  the 
curve  in  but  two  points  each ;  the  general  case  is  omitted,  as  the  solution  in 
such  cases  can  be  rarely  obtained,  and  also  as  the  investigation  is  unsuited  for 
an  elementary  treatise. 

|  This  and  the  next  example  are  taken  from  Cauchy's  Applications  Ge'ome- 
triques  du  Calcul  Infinitesimal^  p.  109. 


272 


Volumes  and  Surfaces  of  Solids. 


then  we  have 

y9  =  b-y/&-(x-a)\     y^b  +  ^/R2- (x-a)\ 
and 

Jyi  i  fy*  x  ibx    / 

zdy  =  -\     xydy  =  —  {y2-y2)  = —  VRi-{x-af. 
y0  eJy0  2C  ° 

Again,  xx  =  a  +  R,    x0  =  a  -  R; 

\Zr*  -(x-dfxdx. 

Now  let    x  -  a  =  R  sin  0,  and  we  get 

- 
2b  R2  C2 
V=~\    co826{a  +  R  sin  B)dB. 

2 

ir  it 

But  '  cos20tf0  =  -,       f  cos20  sin 0^0  =  o, 


v-—ra 

c 


.-.  V 


abR* 


Again,  if  for  the  cylindrical  surface  which  has  for  its 
base  the  circle  we  substitute  a  system  of  four  planes  x  =  x0, 
x  =  X,y  =  y0,y  =  F,  we  get 


=  -(^-V)(F2-^) 
4C 

=  (X-  x0)  (Y-y0) , 


Double  Integration.  273 

in  which  zly  s2>  83,  z4 ,  are  the  ordinates  of  the  four  corner 
points  of  the  portion  of  the  surface  in  question. 

Again,  from  the  well-known  properties  of  the  surface,  in 
order  to  construct  the  hyperbolic  paraboloid  it  is  sufficient 
to  trace  the  gauche  quadrilateral  whose  summits  are  the 
extremities  of  the  ordinates  zl9  z2,  z3,  s4;  then  a  right  line 
moving  on  a  pair  of  opposite  sides  of  this  quadrilateral,  and 
comprised  in  a  plane  parallel  to  the  other  pair,  will  generate 
the  paraboloid  in  question. 

Hence  we  arrive  at  the  following  proposition : — 

Having  traced  a  gauche  quadrilateral  on  the  four  lateral 
faces  of  a  right  prism  standing  on  a  rectangular  base,  if  a 
right  line  move  on  two  opposite  sides  of  this  quadrilateral 
and  be  parallel  to  the  planes  of  the  faces  which  contain  the 
other  two  sides,  then  the  volume  cut  from  the  prism  by  the 
surface  so  generated  is  equal  to  the  product  of  the  area  of 
the  rectangular  base  of  the  prism  by  one-fourth  of  the  sum 
of  the  edges  of  the  prism  between  the  vertices  of  the 
rectangle  and  those  of  the  quadrilateral. 

185.  Double  Integration. — From  the  preceding  Article 
it  is  readily  seen  that  the  double  integral 

J /(a?,  y)dydx 

can  be  represented  geometrically  by  a  volume  ;  and  the  deter- 
mination of  the  double  integral,  when  the  limits  are  given,  is 
the  same  as  the  finding  the  volume  of  a  solid  with  correspond- 
ing limits. 

For  instance,  the  example  in  the  preceding  page  is  equi- 
valent to  finding  the  value  of  the  double  integral 

xydxdy 

taken  for  all  values  of  x  and  y  subject  to  the  condition 

(x-ay+  {y-by-B*<o; 

and  similarly  in  other  cases. 

When  the  limits  of  x  and  y  are  constants,  as  in 


\dydx, 


274  Volumes  and  Surfaces  of  Solids. 

the  double  integral  represents  the  volume  cut  by  the  surface 

2  -/(*,  y) 
from  the  parallelepiped  whose  base  is  the  rectangle  formed 
by  the  lines 

x  =  a,     x  =  a\     y  -b,     y  =  b'. 

It  is  plain  that  in  this  case  the  order  of  integration  is  in- 
different, as  already  seen  in  Art.  115. 

186.  It  is  sometimes  more  convenient  to  refer  the  curve 
RPR'Q  to  polar  co-ordinates,  in  which  case  we  conceive  the 
area  divided  into  infinitesimal  rectangles  of  the  type  rdrdd. 

The  corresponding  parallelepiped  is  represented  by 
zrdrdd,  and  the  expression  for  V  becomes 


zr  dr  dQ, 


taken  between  proper  limits. 

For  instance,  if  the  bounding  surface  be  a  sphere,  whose 
centre  is  the  origin,  we  have 

and  the  equation  becomes 


-JJ^ 


r%  r  dr  dd  ; 

but  j  yV^72  r dr  =  -  1  (a2  -  r2)*. 

Hence,  if  V  denote  the  volume  included  between  the 
sphere  and  the  exterior  surface  of  the  cylinder,  we  shall  have 


r=ijV-»*)«<», 


where  we  suppose  each  radius  of  the  sphere  to  cut   the 
cylinder  in  but  one  point. 

For  example,  let  the  base  of  the  cylinder  be  the  pedal  of 
an  ellipse  whose  major  axis  coincides  with  a  diameter  of  the 
sphere;  then 

r2  =  a2cos20  +  fc2sin20, 

and  F= -Ha2 -62)*  J  sin3 0rf0. 


Double  Integration.  275 

If  this  be  integrated  between  the  limits  o  and  -  we  get 
the  -J-th  of  the  entire  volume ;  hence  the  entire  volume 

V=—  (a2-  b*)K 


Examples. 

I.  A  sphere  is  cut  by  a  right  cylinder,  the  radius  of  whose  base  is  half  that 
of  the  sphere,  and  one  of  whose  edges  passes  through  the  centre  of  the  sphere  ; 
find  the  volume  common  to  both  surfaces. 


27T 


Ans.  ■ ,  a  being  the  radius  of  the  sphere. 

3  9 

2.  If  the  base  of  the  cylinder  be  the  complete  curve  represented  by  the 
equation  r  =  a  cos  nd,  where  n  is  any  integer,  find  the  volume  of  the  solid  be- 
tween the  surface  of  the  sphere  and  the  external  surface  of  the  cylinder. 

187.  It  is  readily  seen,  as  in  Art.  141,  that  the  volume  in- 
cluded within  the  surface  represented  by  the  equation 


*6 


fx    y     z 
V   ~c 


is  abc  x  the  volume  of  the  surface 

F(x,  y,  z)  =  o. 

For,  let  -  =  x\    tb  v\    -  =  2',  and  we  shall  have 
a  be 

zdxdy  =  abcz'  dx'dy, 
and  . •.  /  j zdxdy  =  abc  jj z' dx  dy' ; 

which  proves  the  theorem. 

Hence,  for  example,  the  determination  of  the  volume  of 
an  ellipsoid  is  reduced  to  that  of  a  sphere. 

Again,  if  the  point  (x,  y,  z)  move  along  a  plane,  the  cor- 
responding point  (x',  y\  z')  will  describe  another  plane.  From 
this  property  the  expression  for  the  volume  of  an  ellipsoidal 
cap  (Art.  182)  can  be  immediately  deduced  from  that  of  a 
spherical  cap  (Art.  170). 

[18  a] 


276  Volumes  and  Surfaces  of  Solids. 

In  like  manner  the  volume  included  between  a  cone  en- 
veloping an  ellipsoid  and  the  surface  of  the  ellipsoid  is  reducible  to 
the  corresponding  volume  for  a  sphere. 

1 88.  Quadrature  on  the  Sphere. — We  next  propose 
to  give  a  brief  discussion  of  quadrature  on  a  sphere,  and 
commence  with  the  results  on  the  subject  usually  given  in 
treatises  on  Spherical  Trigonometry.  In  the  first  place, 
since  the  area  of  a  lune  is  to  that  of  the  entire  sphere  as  the 
angle  of  the  lune  to  four  right  angles,  the  area  of  a  lune  of 
angle  A  is  represented  by  zRzA  ;  where  R  is  the  radius  of 
the  sphere,  and  A  is  expressed  in  circular  measure. 

Again,  the  area  of  a  spherical  triangle  ABC  is  expressed 
by  R2  {A  +  B  +  C  -  it)  ;  for,  the  sum  of  the  three  lunes 
exceeds  the  hemisphere  by  twice  the  area  of  the  triangle,  as 
is  easily  seen  from  a  figure. 

Hence,  it  readily  follows  that  the  area  2  of  a  spherical 
polygon  of  n  sides  is  represented  by 

2  =  R2{A  +  B+  C  +  &G.  -  («-2)7rj; 

A,  B,  C,  &c,  being  the  angles  of  the  polygon. 

This  result  admits  of  being  expressed  in  terms  of  the 
sides  of  the  polar  polygon  ;  for,  representing  these  sides  by 
a',  b',  c,  &c,  we  have 

A  =  tt  -  a\     B  =  7r  -  b',  &c, 
and  consequently 

S  =  i22{27r  -  (a'  +  V  +  c'  +&c.)}. 
Or,  denoting  the  perimeter  of  the  polar  figure  by  S, 

S  +  BS  =  2irR\  (6) 

This  proof  is  perfectly  general,  and  holds  in  the  limit, 
when  the  polygon  becomes  any  curve ;  and,  accordingly,  the 
area  bounded  by  any  closed  spherical  curve  is  connected  with 
the  perimeter  of  its  polar  curve  by  the  relation  (6). 

Again,  the  spherical  area  bounded  by  a  lesser  circle 
(Art.  170)  admits  of  a  simple  expression.  If  p  denote  the 
circular  radius  of  the  circle,  or  the  arc  from  its  pole  to  its 
circumference,  the  area  in  question  is  represented  by 

2ttR2(i  -  cos/o) ; 


Quadrature  on  the  Sphere.  277 

for  (see  fig.  Art.  1 70)  we  have 

AN=AC-  CN=R(i  -cosP). 

This  result  also  follows  immediately  as  a  simple  case  of 
equation  (6). 

Again,  the  area  bounded  by  the  lesser  circle  and  by  two 
arcs  drawn  to  its  pole  is  plainly  represented  by 

M2a(l  -  COS/o), 

where  a  is  the  circular  measure  of  the  angle  between  the  arc?. 

We  can  now  find  an  expression  for  the  area  bounded  by 
any  closed  curve  on  a  sphere ;  for 
the  position  of  any  point  P  on  the 
surface  can  be  expressed  by  means 
of  the  arc  OP  drawn  to  a  fixed 
point,  and  of  the  angle  POX 
between  this  arc  and  a  fixed  arc 
through  0.  These  are  called  the 
polar  co-ordinates  of  the  point,  and 
are  analogous  to  ordinary  polar 
co-ordinates  on  a  plane.  Pi~  46 

Now,  let  OP  =  p,  and  POX  -  to ; 
then  any  curve  on  the  sphere  maybe  supposed  to  be  expressed 
by  a  relation  between  p  and  w. 

Again,  suppose  OQ  to  represent  an  infinitely  near  vector, 
and  draw  PR  perpendicular  to  OP;  then,  neglecting  in 
the  limit  the  area  PQR,  the  elementary  area  OPQ  by  the 
preceding  is  represented  by 

i22(i  -  cos  p)d(o. 

Hence  the   area  bounded  by   two  vectors   from    0  is 

expressed  by  the  integral  P?     (1  -  cosp)c?w,  taken  between 

suitable  limits. 

If  the  curve  be  closed,  the  entire  superficial  area  becomes 


(1  -  cos  p)du). 

0 


The  value  of  cos  p  in  terms  of  w  is  to  be  determined  in 
each  case  by  means  of  the  equation  of  the  bounding  curve. 


278  Volumes  and  Surfaces  of  Solids. 

J2»r 
cos  p  dto  obviously  represents  the  area 

included  between  the  closed  curve  and  the  great  circle  which 
has  0  for  its  pole. 

The  length  of  the  curve  can  also  be  represented  by  a 
definite  integral ;  for,  regarding  PRQ  as  ultimately  a  right- 
angled  triangle,  we  have  in  the  limit, 

PQ2  =  PR2  +  RQ2  :  also  PR  =  sinptfw. 
Hence  ds2  =  dr2  +  Bm2p  du2, 

or  ds  =  dto  J  sin2/>  +  (-/-)> 

.-.  .-J«foJrinV+(g 

Again,  it  is  manifest  from  (6)  that  the  determination  of 
the  length  of  any  spherical  curve  is  reducible  to  finding  the 
area  of  its  polar  curve,  and  vice  versa. 

Examples. 

i.  Find  the  area  of  the  portion  of  the  surface  of  a  sphere  which  is  inter- 
cepted hy  a  right  cylinder,  one  of  whose  edges  passes  through  the  centre  of  the 
sphere,  and  the  radius  of  whose  hase  is  half  that  of  the  sphere. 

Here,  the  equation  of  the  hase  may  be  written  in  the  form  r  =  £  sin  w, 
-R  being  the  radius  of  the  sphere,  and  a  being  measured  from  the  tangent  to  the 
circular  base. 

Again,  from  the  sphere  we  have  r  =  B  sin  p ;  .•.  p  =  a  is  the  equation  of 
the  curve  of  intersection  of  the  sphere  and  the  cylinder ;  hence  the  area  in 
question  is 

5 

2i22f    (I- cos  o»)dw  =  ill*  (--i\. 

This  being  doubled  gives  the  whole  intercepted  area  =  2tr  JR?  —  4-R2. 

This  is  the  celebrated  Florentine  enigma,  proposed  by  Vincent  Viviani  as  a 
challenge  to  the  Mathematicians  of  his  time,  in  the  following  form  : — "  Inter 
venerabilia  olim  Graecise  monumenta  extat  adhuc,  perpetuo  quidem  duraturum, 
Templum  augustissimum  ichnographia  circulari  Almse  Geometriae  dicatum,  quod 
Testudine  intus  perf  ecte  hemisphserica  operitur :  sed  in  hac  f  enestrarum  quatuor 
aequales  areae  (circum  ac  supra  basin  hemisphserae  ipsius  dispositarum)  tali  con- 
figuratione,  amplitudine,  tantaque  industria,  ac  ingenii  acumine  sunt  exstructa?, 


Quadrature  of  Surfaces.  279 

ut  his  detractis,  superstes  curva  Testudinis  superficies,  pretioso  opere  musivo 
ornata,  Tetragonismi  vere  geometricisit  capax." — Acta  JEruditorum,  Leipsic,  1692. 
[See  Montucla,  Histoire  des  Mathematiques,  tome  ii.,  p.  94.] 

In  general,  if  r  =/(«)  be  the  equation  of  the  base  of  a  cylinder,  it  is  easily- 
seen  that  the  equation  of  the  curve  of  its  intersection  with  the  sphere  may  be 
written  in  the  form  R  sin  p  =/(«). 

For  example,  let  the  diameter  of  the  right  cylinder  be  less  than  half  that 
of  the  sphere ;  then  writing  the  equation  of  the  base  in  the  form  r  =  a  sin  &>, 
where  a  is  the  diameter  of  the  section,  we  get  R  sin  p  =  a  sin  ca,  or  sin  p  =  k  sin  a> 
(where  k  is  <  1),  as  the  equation  of  the  curve  of  intersection  of  the  sphere  and 
the  cylinder. 

Hence  the  intercepted  area  is  denoted  by 

it  tr 

2R*  J    (1  -  s/i  -K2sin2o>)rf«  =  ttR2  -  2R*  J    v^1  -/c2suvWw. 

Hence  the  area  in  question  depends  on  the  rectification  of  an  ellipse. 

2.  Find  the  area  of  the  portion  of  the  surface  of  the  cylinder  intercepted  by 
the  sphere,  in  the  preceding. 

Here  the  area  in  question  is  easily  seen  to  be  represented  by  2  J  zds,  where 
ds  denotes  the  element  of  the  curve  which  forms  the  base,  corresponding  to  the 
edge  z. 

Now  (1),  when  the  diameter  of  the  base  is  equal  to  the  radius  of  the  sphere, 
we  have 

z  =  R  cos  w,  and  ds  =  Rdw  ; 

IT 

.-.  area  in  question  =  2.R2  I  cos  wdu  =  4^  ;  i.e.  the  square  of  the  diameter  of 
the  sphere. 

2.  When  the  diameter  is  less  than  the  radius  of  the  sphere, 

2  J  zds  =  2a     \/ R2  -  a2  sin2«<f«  =  2aR  \  \/ 1  —  K2sin2o>  do> ;  .-.  &c. 

189.  Quadrature  of  Surfaces. — In  seeking  the  area 
of  a  portion  of  any  surface  we  regard  it  as  the  limit  of  a 
number  of  infinitely  small  elements,  each  of  which  is  con- 
sidered as  a  portion  of  a  plane  which  is  ultimately  a  tangent 
plane  to  the  surface.  Now  let  dS  denote  such  an  element  of 
the  superficial  area,  and  da  its  projection  on  a  fixed  plane 
which  makes  the  angle  9  with  the  plane  of  the  element ;  then, 
from  elementary  geometry,  we  shall  have 

da  =  cos  OdS,  or  dS  =  sec  6  cl 'a. 


Hence  S 


sec  9  da, 


taken  between  suitable  limits. 


280  Volumes  and  Surfaces  of  Solids. 

The  applications  of  this  formula  usually  involve  double 
integration,  and  are  generally  very  complicated  ;  there  is, 
however,  one  mode  by  which  the  determination  of  the  area  of 
a  portion  of  a  surface  can  be  reduced  to  a  single  integration, 
and  by  whose  aid  its  value  can  in  some  cases  be  found  ;  viz., 
by  supposing  the  surface  divided  into  zones  by  a  system*  of 
curves  along  each  of  which  the  angle  0  between  the  tangent 
plane  and  a  fixed  plane  is  constant ;  then,  if  dS  denote  the 
superficial  area  of  the  zone  between  the  two  infinitely  near 
curves  corresponding  to  the  angles  6  and  9  +  dd  ;  and,  if  dA 
be  the  projection  of  this  area  on  the  fixed  plane,  we  shall 
have  dS  =  sec  OdA. 

If  we  suppose  the  surface  referred  to  a  rectangular 
system  of  axes,  the  fixed  plane  being  that  of  xy  ;  and 
.adopting  the  usual  notation,  if  we  take  A,  /*,  v  as  the  direction 
angles  of  the  normal  at  any  point  on  the  surface,  we  get 
for  dS,  the  area  of  the  zone  between  the  curves  corresponding 
to  v  and  v  +  dv,  the  equation 

dS  =  sec  vdA, 

where  A  denotes  the  area  of  the  projection  on  the  plane  of 
ay  of  the  closed  curve  defined  by  the  equation  v  =  constant. 

Now  whenever  we  can  express  the  area  A  in  terms  of  v 
and  constants,  then  the  area  of  a  portion  of  the  surface, 
bounded  by  two  curves  of  the  system  in  question,  is  reducible 
to  a  single  integration. 

The  most  important  applications  of  this  method  are 
furnished  by  surfaces  of  the  second  degree,  to  which  we 
proceed  to  apply  it,  commencing  with  the  paraboloid. 

1 90.  Quadrature  of  the  Paraboloid. — Writing  the 
equation  of  the  surface  in  the  form 

x1      y2 

p     q 


*  This  method  has  heen  employed  in  a  more  or  less  modified  form  by 
M.  Catalan,  Ziouville,  tome  iv.,  p.  323,  by  Mr.  Jellett,  Camb.  and  Dub.  Math. 
Journal,  vol.  i.,  as  also  bv  other  writers.  The  curves  employed  are  called 
parallel  curveshjM.  Lebesgne,  Ziouville,  tome  xi.,  p.  332,  and  Curven  isokliner 
Kormalen,  by  Dr.  JSchlomilch. 


Quadrature  of  the  Paraboloid.  281 

the  equation  of  the  tangent  plane  at  the  point  (x,  y,  z)  is 
xX     yY 

p       q 

where  X,  Y,  Z  are  the  co-ordinates  of  any  point  on  the  plane. 
Comparing  this  with  the  equation 

X  cos  X  +  Y  cos  fj.  +  Z  cos  v  =  P, 

X  If 

we  get  cos  X  =  —  cos  v,    cos  u  =  -  -  cos  v  : 

P  q 

substituting  in  the  identical  equation 

COS2X  +  COS2jU  +  cos2v  =  I, 

x2     y2 
we  get  — +  — =  tan2v.  (7) 

p2      q~  ' 

Consequently  the  curve  along  which  the  tangent  plane 
makes  the  angle  v  with  the  tangent  plane  at  the  vertex  is 
projected  on  that  plane  into  the  ellipse 

3  +  £  -  tanV 
p2      q2 

The  area  A  of  this  ellipse  is  Trpqk&tfv  ;  accordingly,  we 
have 

dA  =  irpqd  (tan2  v)  ; 

.*.  dS  =  irpq  sec  vc?(tan2  v)  =  irpq  sec  vc/(sec2v) ; 

hence  the  area  of  the  paraboloidal  cap  bounded  by  the  curve 
v  =  a  is 


npq 


secvd(seG2v)  =  %irpq(seGza  -  1). 


Also  the  area  of  the  belt*  between  the  curves 

v  =  a  and  v  =  a  is  -§77^ (sec3 a'  -  sec3 a).  (8) 

*  This  form  for  the  quadrature  of  a  paraboloid  is,  I  believe,  due  to  Mr.  Jellett: 
see  Camb.  and  Dub.  Math.  Journal,  vol.  i.  p.  65.  The  proof  given  above  is  in 
a  great  measure  taken  from  Mr.  Allman's  paper  in  the  Quarterly  Journal,  already 
referred  to. 


282  Volumes  and  Surfaces  of  Solids. 

191.  Quadrature  of  the    Ellipsoid. — Proceeding  in 
like  manner  to  the  ellipsoid 

x2     y2     z2 

a2      b2     c2 

the  equation  of  the  tangent  plane  at  the  point  (x,  y,  z)  is 

Xx      Yy     Zz 


Hence, 

comparing 

with  the 

equation 

we 

get 

XcosX 

c2 

cos  X  =  — • 

a2 

+  Yoosfx 

X 

—  cos  V, 

1 

+  Z  cos  v  ■■ 

c2 

COS  fl  =  -2 

y 

-  COS  V, 

z 

Hence,  we  have 

c*  (x2     v2\ 
cos2i/—  —  +  — )  =  cos2X  +  cos2u  =  sin'v ; 
z2  \a*     07 


v  J  =  sin2  v. 


X2       V2  z2 

or,  substituting  1 -  j-2  f or  -, 

-  (  a2  sin2 v  +  c2  cos2 v  ]  +  tt  I  o2  sin2v  +  c2  cos2 

«4V  ;    &4\ 

This  shows  that  the  projection  on  the  plane  of  xy  of 
a  curve  along  which  v  =  constant  is  an  ellipse. 
Again  the  area  A  of  this  ellipse  is 

tto2  b2  sin2  v 

(a2  sin2v  +  e2  co&2v)*(b2  sin2v  +  c2  cos^)*' 

and   accordingly,   the  area  dA   of  the  elementary  annulus 
between  two  consecutive  ellipses  is 

dv  \  (a2  sin2  v  +  c2  cos2  v)*  (b2  sin2  v  +  c2  cos2 v)£) 

The    corresponding  elementary  ellipsoidal    zone   dS  is 
represented  by 

ira2b2  d  (  sin2v  )  . 

av> 


cos  v  tfv  \  {a2  sin2v  +  c2  cos2  v)*  (b2  sin2  v  +  c2  cos2  v)*J 


Quadrature  of  the  Ellipsoid.  283 

Now,  if  8  denote  the  superficial  area*  between  two 
curves  corresponding  to  v  =  a  and  v  =  a',  after  one  or  two 
reductions,  it  is  easily  seen  that 


where 


£  =  7ra2&V  (/+/'),  (9) 

J'a'  sin  v  dv 

a  (b2  sin2  v  +  c2  cos2  v)1  (a2  sin2v  +  c2cos2  vj*' 

f*  sin  v  dv 

'  J  a  (a2  sin2  v  +  c2  cos2  v)%  (b2  sin2  v  +  c2  cos2v)2' 

It  is  easily  shown  that  the  former  of  these  integrals  is 
represented  by  an  arc  of  an  ellipse,  and  the  latter  by  an  arc 
of  a  hyperbola  ;  it  being  assumed  that  a  >  b  >  c. 

For,  assuming  a2  -  c2  =  «V,  and  b2  -  c2  =  b2e'2,  and 
making  cos  v  =  x,  we  get 


1      ab" 


dx 


T     ("  cos  a 

r'=—  — 

«3&Jcosa    (I 


eosa(l-e,2^(l-^y 


(i  -  e2x2)*(i  -e'2x2)h' 

Again,  let  ex  =  sin  6  in  the  former  integral,  and  ex  =  sin  9 
in  the  latter,  and  we  get 


<$ 


-e'2sin20)^ 

r     g  f  dB 

azb)(e'2-e2mrQf 


Now,  since  e  >  e',  the  former  integral  represents  an 
arc  of  an  ellipse,  and  the  latter  an  arc  of  a  hyperbola.  (See 
Ex.  19,  p.  249). 


*  This  form  for  the  quadrature  of  an  ellipsoid  is  given  by  Mr.  Jellett  in 
the  memoir  already  referred  to.  He  has  also  shown  that  the  ellipse  and  the 
hyperbola  in  question  are  the  focal  conies  of  the  reciprocal  ellipsoid ;  a  result 
which  can  be  easily  arrived  at  from  the  forms  of  I  and  i"'  given  above. 

For  application  to  the  hyperboloid,  and  further  development  of  these  results, 
the  student  is  referred  to  Mr.  Jellett' s  memoir. 


284  Volumes  and  Surfaces  of  Solids. 

192.  Integration  over  a  Closed  Surface. — We  shall 
conclude  this  Chapter  with  the  consideration  of  some  general 
formulae  in  double  integration  relative  to  any  closed  surface. 
We  commence  by  adopting  the  same  notation  as  in  Art.  189, 
where  A,  /u,  v  are  taken  as  the  angles  which  the  exterior 
normal  at  the  element  dS  makes  with  the  positive  directions 
of  the  axes  of  x,  y,  2,  respectively. 

Again,  let  each  element  of  the  surface  be  projected  on 
the  plane  of  xy,  and  suppose*  for  simplicity  that  each  z  ordi- 
nate meets  the  surface  in  but  two  points :  then,  if  the  indefi- 
nitely small  cylinder  standing  on  any  element  dA  in  the 
plane  of  xy  intersects  the  surface  in  the  two  elementary  por- 
tions dSi  and  dS2  (where  dSi  is  the  upper,  and  dS2  the  lower 
element),  and  if  vx  and  v2  be  the  corresponding  values  of  v,  it 
is  plain  that  vx  is  an  acute,  and  v2  an  obtuse  angle,  and  we 
have 

dA  =  cos  vidSi  =  -  cos  v2dS2. 

Hence,  if  we  take  into  account  all  the  elements  of  the  surface, 
attending  to  the  sign  of  cos  v,  we  shall  have 

jJQOSvdS  =  o. 

In  like  manner  we  get 

Jj cos  XdS  =  o,  and  jj cos  fi dS  =  o ; 

the  integrals  extending  in  each  case  over  the  whole  of  the 
closed  curve 

These  formulae  are  comprised  in  the  equation 

J7  (a  cos  A  +  j3  cosju  +  7  cos  v)dS  =  o.  (10) 

Again,  if  zx  and  z2  be  the  values  of  z  corresponding  to  the 
element  dA,  then,  denoting  by  dVthe  element  of  volume 
standing  on  dA  and  intercepted  by  the  surface,  we  plainly 
have 

dV=  (zi  -  z2)dA  =  z\dSi  cos  vi  +  z2dS2  cos  v2, 


*  It  it  easily  seen  that  this  and  the  following  demonstrations  are  perfectly 
general,  inasmuch  as  each  ordinate  must  meet  a  closed  surface  in  an  even  number 
of  points,  which  may  he  considered  in  pairs. 


Integration  over  a  Closed  Surface.  285 

and  the  sum  of  all  such  elements,  that  is,  the  whole  volume, 
is  evidently  represented  by 

jjz  cos  vdS. 

Hence,  denoting  the  whole  volume  by  V,  we  have 

V=  jjx  cosXdS  =  jjy  cos  fid8  =  jj  zcosvdS; 

the  integrals,   as  before,   being   extended   over  the   entire 
surface. 

Again,  it  is  easily  seen  that  we  have 

jjx  cos  vdS  =  o,    jjy  cos  vdS  =  o,    jj  x  cos /j.dS  =  o, 

jjy  cosX  dS  =  o,    jjz  cos \dS=  o,    j  j  z  cos  fidS  =  o. 

For,  as  in  the  first  case,  it  readily  appears  that  the  elements 
are  equal  and  opposite  in  pairs  in  each  of  these  integrals. 
These  results  are  comprised  in  the  equation 

jj(ax  +  j3y  +  yz)  (a  cos  X  +  j3'  cos  ju  +  y  cos  v)  dS 

=  (aa'  +  /3j3'  +  7y')F.      (ll) 
For  a  like  reason,  we  have 
j  j  xy  cos  v  d8  =  o,    jjzx  cos  fidS  =  o,    jjyz  cosXdS  =  o. 
Also         JJV  cos  vdS  =  o,    jj  x7  cos  fidS  =  o,  &c. 
Next,  let  us  consider  the  integral 
jjxz  cos  vdS. 

This  integral  is  equivalent  to  jj  xdV;  consequently,  if 
x,  y9  s,  be  the  co-ordinates  of  the  centre  of  gravity  of  the 
enclosed  volume  F,  we  get  jj  xz  cos  vdS  =  jjxd  V  =  xV;  in 
like  manner  jj  xz  cos  Xd8  =zV. 

Again,  the  integral 

jjz2  cos  vdS 
consists  of  elements  of  the  form  (si2  -  zi)  dA  ;  but 
(zi*  -  z22)  dA  =  (zl  +  s2)  (si  -  S2)  dA 
=  (zi  +  z2)dV. 


286  Volumes  and  Surfaces  of  Solids. 


But  the  s  ordinate  of  the  centre  of  gravity  of  dV  is 
plainly  — 2,  and  consequently 

I L2 cos vdS=  2  [ [Z-±^ dV  =  2zV. 

In  like  manner  it  can  be  shown  that 

jjx2  cosXdS  =  2xV,    jjy2cosfidS  =  2yV. 

Accordingly  we  have 

Vx  =  i  jjx2  cos  A  dS  =  H xy  cos  fidS  =  jjxz  cos  vdS, 

Vy  =  jjyx  cosXdS  =  %  jjy2  cos  fjidS  =  jjyzcosvdS, 

Vz  =  j  j  zx  cos  XdS  =  jjzy  cospdS  =  ^l\z2  cos  vdS. 

193.  Expression  for  Volume  of  a  Closed  Surface. 

— Next,  if  we  suppose  a  cone  described  with  its  vertex 
at  the  origin  0,  and  standing  on  the  elementary  base  dS, 
its  volume  is  represented  (Art.  1 69)  by  ^pdS,  where  p  is  the 
length  of  the  perpendicular  drawn  from  0  to  the  tangent 
plane  at  the  point. 

Also,  if  r  be  the  distance  of  0  from  the  point,  and  y  the 
angle  which  r  makes  with  the  internal  normal,  we  have 
p  =  r  cos  7. 

Hence  the  elementary  volume  is  equal  to  -J-  r  cos  y  dS,  and 
it  is  easily  seen  that  if  we  integrate  over  the  entire  surface, 
the  enclosed  volume  is  represented  by 

Ifjjr  cos ydS. 

1 94.  Again,  if  we  suppose  a  sphere  of  unit  radius  described 
with  0  as  centre,  and  if  da)  represent  the  superficial  portion 
of  this  sphere  intercepted  by  the  elementary  cone  standing  on 
dS,  then  it  is  easily  seen  that  cos  ydS  =  i^du  ; 

.        COS  y  dS 
.'.  dio  = — . 

r2 

Now  if  0  be  inside  the  closed  surface,  and  the  integral 
be  extended  over  the  entire  surface,  it  is  plain  that  jj  du>  =  47r, 
being  the  surface  of  the  sphere  of  radius  unity ; 

COSydS 


47T. 


Expression  for  Volume  of  a  Closed  Surface.  287 

Again,  if  0  be  outside  the  surface,  the  cone  will  cut  the 
surface  in  an  even  number  of  elements,  for  which  the  values 
of  cos  y  will  be  alternately  positive  and  negative,  and,  the 
corresponding  elements  of  the  integral  being  equal  but  with 
opposite  signs,  their  sum  is  equal  to  zero,  and  we  shall  have 


ff 


cos  7  dS 
i —  =  o. 


If  0  be  situated  on  the  surface,  it  follows  in  like  manner 
that 

Hence,  we  conclude  that 

* 

— ^-dS  =  47r,  27r,  or  o,  (12) 


ff 


according  as  the  origin  is  inside,  on,  or  outside  the  surface. 

The  multiple  integrals  introduced  into  this  and  the  two 
preceding  Articles  are  principally  due  to  Gauss. 

The  student  will  find  some  important  applications  of 
this  method  in  Bertrand's  Calc.  Int.,  §§437,  455,  456, 
476,  &o. 


288  Examples. 


Examples. 

i.  A  sphere  of  15  feet  radius  is  cut  by  two  parallel  planes  at  distances  of 
3  and  7  feet  from  its  centre ;  find  the  superficial  area  of  the  portion  of  the  sur- 
face included  between  the  planes  approximately.  Ans.  376.9908  sq.  feet. 

2.  Being  given  the  slant  height  of  a  right  cone,  find  the  cosine  of  half  its 
vertical  angle  when  its  volume  is  a  maximum.  .  1 

Ans.    — -, 

3.  Prove  that  the  volume  of  a  truncated  cone  of  height  h  is  represented  by 

—  (JP  +  JRr  +  H), 

where  H  and  r  are  the  radii  of  its  two  bases. 

4.  A  cone  is  circumscribed  to  a  sphere  of  radius  i?,  the  vertex  of  the  cone 
being  at  the  distance  D  from  the  centre ;  find  the  ratio  of  the  superficial  area  of 
the  cone  to  that  of  the  sphere.  IP  —  Ri 

Ans.  — — — -. 
4DE 

5.  Two  spheres,  A  and  B,  have  for  radii  9  feet  and  40  feet ;  the  superficial 
area  of  a  third  sphere  C  is  equal  to  the  sum  of  the  areas  of  A  and  B ;  calculate 
the  excess,  in  cubic  feet,  of  the  volume  of  C  over  the  sum  of  the  volumes  of  A 
and  J?.  Ans.  17558. 

6.  If  any  arc  of  a  plane  curve  revolve  successively  round  two  parallel  axes, 
show  that  the  difference  of  the  surfaces  generated  is  equal  to  the  product  of  the 
length  of  the  arc  into  the  circumference  of  the  circle  described  by  any  point  on 
either  axis  turning  round  the  other. 

If  the  axes  of  revolution  lie  at  opposite  sides  of  the  curve,  the  sum  of  the 
surfaces  must  be  taken  instead  of  the  difference. 

7.  Find,  in  terms  of  the  sides,  the  volume  of  the  solid  generated  by  the 
complete  revolution  of  a  triangle  round  its  side  c. 

Ans.  ^s(s-a)(s-b)(s-e) 
'    3  <> 

8.  Apply  Guldin's  theorem  to  determine  the  distance,  from  the  centre,  of  the 
centre  of  gravity,  ( 1)  of  a  semicircular  area  ;  (2)  of  a  semicircular  arc. 

4«  2a 

Ans.    1   -,      2  — . 
3ir  ir 

9.  If  a  triangle  revolve  round  any  external  axis,  lying  in  its  plane,  find  an 
expression  for  the  area  of  the  surface  generated  in  a  complete  revolution. 

10.  Prove  that  the  volume  cut  from  the  surface 

z»  =  Ax2  +  By2 

by  any  plane  parallel  to  that  of  xy,  is th  part  of  the  cylinder  standing  on 

the  plane  section,  and  terminated  by  the  plane  of  xy. 


Examples.  289 

ii.  A  cone  is  circumscribed  to  a  sphere  of  23  feet  radius,  the  vertex  of  the 
oone  being  265  feet  distant  from  the  centre  of  the  sphere  ;  find  the  ratio  of  the 
superficial  area  of  the  cone  to  that  of  the  sphere. 

12,  The  axis  of  a  right  circular  cylinder  passes  through  the  centre  of  a 
sphere  ;  find  the  volume  of  the  solid  included  between  the  concave  surface  of  the 
sphere  and  the  convex  surface  of  the  cylinder. 

Ans.  ^— ,  where  c  is  the  length  of  the  portion  of  any  edge  of  the  cylinder 
6 
intercepted  by  the  sphere. 

This  question  is  the  same  as  that  of  finding  the  volume  of  the  solid  generated 
by  the  segment  of  a  circle  cut  off  by  any  chord,  in  a  revolution  round  the 
diameter  parallel  to  the  chord. 

13.  Find  the  volume  of  the  solid  generated  by  the  revolution  of  an  arc  of  a 

(2a2  +  c2)  sin  a 


circle  round  its  chord.  Ans.  2-rra 

3 

c 
where  a  =  radius,  c  =  distance  of  chord  from  centre,  and  cos  a  =  -. 

a 

In  this  we  suppose  the  arc  less  than  a  semicircle  :  the  modification  when  it 
is  greater  is  easily  seen. 

14.  If  the  ellipsoid  of  revolution, 
and  the  hyperboloid 


a2v2 


tf-b2 


be  cut  by  two  planes  perpendicular  to  the  axis  of  revolution,  prove  that  the 
zones  intercepted  on  the  two  surfaces  are  of  equal  area. 

15.  Find  the  entire  volume  bounded  by  the  positive  sides  of  the  three  co- 
ordinate planes,  and 


(-lHf)H)1- 


.        abe 
Ans.  — . 
90 


16.  Find  the  volume  of  the  surface  generated  by  the  revolution  of  an  arc  of 
a  parabola  round  its  chord ;  the  chord  being  perpendicular  to  the  axis  of  the 

curve. 

g 
Ans.  — irb2c,  where  c  is  the  length  of  the  chord,  and  b  the  intercept  made 

by  it  on  the  diameter  of  the  parabola  passing  through  the  middle  point  of  the 
chord. 

17.  A  sphere  of  radius  r  is  cut  by  a  plane  at  distance  d  from  the  centre  ;  find 
the  difference  of  the  volumes  of  the  two  cones  having  as  a  common  base  the 
circle  in  which  the  plane  cuts  the  sphere,  and  whose  vertices  are  the  opposite 
ends  of  the  diameter  perpendicular  to  the  cutting  plane. 

Ans.  %ird  (r»  -  d2). 
[19] 


290  Examples. 

1 8.  Find  the  area  of  a  spherical  triangle ;  and  prove  that  if  a  curve  traced 
on  a  sphere  have  for  its  equation  sin  A.  =  f(l),  \  denoting  latitude,  and  /  longi- 
tude, the  area  between  the  curve  and  the  equator  ■  jf(l)dl. 

19.  Show  that  the  volume  contained  between  the  surface  of  a  hyperboloid 
of  one  sheet,  its  asymptotic  cone,  and  two  planes  parallel  to  that  of  the  real 
axes,  is  proportional  to  the  distance  between  those  planes. 


20.  Find  the  entire  volume  of  the  surface 


(=)•*(?)'♦  G) 


z  \  i  .       Airabc 

-  I.  Ans. . 

5  •  7 


31.  The  vertex  of  a  cone  of  the  second  degree  is  in  the  surface  of  a  sphere, 
and  its  internal  axis  is  the  diameter  passing  through  its  vertex  ;  find  the  volume 
of  the  portion  of  the  sphere  intercepted  within  the  cone. 

22.  Prove  that  the  volume  of  the  portion  of  a  cylinder  intercepted  between 
any  two  planes  is  equal  to  the  product  of  the  area  of  a  perpendicular  section 
into  the  distance  between  the  centres  of  gravity  of  the  areas  of  the  bounding 
sections. 

23.  If  A  be  the  area  of  the  section  of  any  surface  made  by  the  plane  of  xy, 
prov*,  us  in  Art.  192,  that 

A  =  fjcosi/dS, 

the  integral  being  extended  through  the  portion  of  tbe  surface  which  lies  above 
the  plwae  of  xy. 

24.  If  a  right  cone  stand  on  an  ellipse,  prove  that  its  volume  is  represented 
by 

-  (OA  .  OA')%  sin2a  cos  a ; 

where  0  is  the  vertex  of  the  cone,  A  and  A'  the  extremities  of  the  major  axis 
of  the  ellipse,  and  a  is  the  semi -angle  of  the  cone. 

25.  In  the  same  case  prove  that  the  superficial  area  of  the  cone  ia 

-  (OA  +  OA')  (OA  .  OA')l  sin  a. 


291     ) 


CHAPTER  X. 

INTEGRALS     OF     INERTIA. 

195.  Integrals  of  Inertia. — The  following  integrals  are 
of  such  frequent  occurrence  in  mechanical  investigations, 
that  it  is  proposed  to  give  a  brief  discussion  of  them  in  this 
Chapter. 

If  each  element  of  the  mass  of  any  solid  body  be  supposed 
to  be  multiplied  by  the  square  of  its  distance  from  any  fixed 
right  line,  and  the  sum  extended  throughout  every  element 
of  the  body,  the  quantity  thus  obtained  is  called  the  moment 
of  inertia  of  the  body  with  respect  to  the  fixed  line  or  axis. 

Hence,  denoting  the  element  of  mass  by  dm,  its  distance 

from  the  axis  by  p,  and  the  moment  of  inertia  by  1,  we  have 

I=^p2dm.  (1) 

In  like  manner,  if  each  element  of  mass  of  a  body  be 
multiplied  by  the  square  of  its  distance  from  a  plane,  the 
sum  of  such  products  is  called  the  moment  of  inertia  of  the 
body  relative  to  the  plane. 

If  the  system  be  referred  to  rectangular  axes   of  co- 
ordinates, then  the  expression  for  the  moment  of  inertia 
relative  to  the  axis  of  z  is  obviously  represented  by 
2  (x2  +  y2)  dm. 

Similarly,  the  moments  of  inertia  relative  to  the  axes  of 
x  and  y  are  represented  by  S  (y2  +  z2)  dm  and  S  (x2  +  z3)  dm, 
respectively. 

Again,  the  quantities  '2,x2dm,  ^y2dm,  ^z2dm,  are  the 
moments  of  inertia  of  the  body  with  respect  to  the  planes 
of  yz,  xz,  and  xy,  respectively.  Also  the  quantities  ^xydm, 
^zxdm,  ILyzdm,  are  called  the  products  of  inertia  relative  to 
the  same  system  of  co-ordinate  axes. 

In  like  manner  the  moment  of  inertia  of  the  body  icith 
reference  to  a  point  is  ^,r2dm,  where  r  denotes  the  distance  of 
the  element  dm  from  the  point.  Thus  the  moment  of  inertia 
relative  to  the  origin  is  2  (x2  +  y2  +  z2)  dm. 

[19  a] 


292  Integrals  of  Inertia. 

196.  Moments  of  Inertia  relative  to  Parallel 
Axes,  or  Planes. — The  following  result  is  of  fundamental 
importance  : — The  moment  of  inertia  of  a  body  with  respect  to 
any  axis  exceeds  its  moment  of  inertia  with  respect  to  a  parallel 
axis  drawn  through  its  centre  of  gravity,  by  the  product  of  the 
mass  of  the  body  into  the  square  of  the  distance  between  the 
parallel  axes. 

For,  let  /  be  the  moment  of  inertia  relative  to  the  axis 
through  the  centre  of  gravity,  I'  that  for  the  parallel  axis, 
M  the  mass  of  the  body,  and  a  the  distance  between  the  axes. 

Then,  taking  the  centre  of  gravity  as  origin,  the  fixed 
axis  through  it  as  the  axis  of  2,  and  the  plane  through  the 
parallel  axes  for  that  of  zx,  we  shall  have 

7=  2(z*  +  y2]dm,     I'  =  2{(«  +  a)2  +  y2}dm. 

Hence  T  -  I  =  2a^xdm  +  a2 "2 dm  =  a2M, 

since  Sxdm  =  o  as  the  centre  of  gravity  is  at  the  origin ; 

.-./'  =  /+  a2  M.  (2) 

Consequently,  the  moment  of  inertia  of  a  body  relative  to 
any  axis  can  be  found  when  that  for  the  parallel  axis  through 
its  centre  of  gravity  is  known. 

Also,  the  moments  of  inertia  of  a  body  are  the  same  for 
all  parallel  axes  situated  at  the  same  distance  from  its  centre 
of  gravity. 

Again,  it  may  be  observed  that  of  all  parallel  axes  that 
which  passes  through  the  centre  of  gravity  of  a  body  has  the 
least  moment  of  inertia. 

It  is  also  apparent  that  the  same  theorem  holds  if  the 
moments  of  inertia  be  taken  with  respect  to  parallel  planes, 
instead  of  parallel  axes. 

A  similar  property  also  connects  the  moment  of  inertia 
relative  to  any  point  with  that  relative  to  the  centre  of 
gravity  of  the  body. 

In  finding  the  moment  of  inertia  of  a  body  relative  to 
any  axis,  we  usually  suppose  the  body  divided  into  a  system 
of  indefinitely  thin  plates,  or  lamince,  by  a  system  of  planes 
perpendicular  to  the  axis ;  then,  when  the  moment  of  inertia 
is  determined  for  a  lamina,  we  seek  by  integration  to  find 
that  of  the  entire  body. 


Radius  of  Gyration.  293 

197.  Radius  of  Gyration. — If  k  denote  the  distance 
from  an  axis  at  which  the  entire  mass  of  a  body  should  be 
concentrated  that  its  moment  of  inertia  relative  to  the  axis 
may  remain  unaltered,  we  shall  have 

Mk2  =  I=-2p2dm.  (3) 

The  length  k  is  called  the  radius  of  gyration  of  the  body 
with  respect  to  the  fixed  axis. 

In  homogeneous  bodies,  which  shall  be  here  treated  of 
principally,  since  the  mass  of  any  part  varies  directly  as  its 
volume,  the  preceding  equation  may  be  written  in  the  form 

where  dV  denotes  the  element  of  volume,  and  F'the  entire 
volume  of  the  body. 

Hence,  in  homogeneous  bodies,  the  value  of  k  is  indepen- 
dent of  the  density  of  the  body,  and  depends  only  on  its  form. 

We  shall  in  our  investigations  represent  the  moment  of 
inertia  in  the  form  j     ■**■„ . 

and,  it  is  plain  that  in  its  determination  for  homogeneous 
bodies  we  may  take  the  element  of  volume  for  the  element  of  mass, 
and  the  total  volume  of  the  body  instead  of  its  mass. 

Also,  in  finding  the  moment  of  inertia  of  a  lamina,  since  its 
radius  of  gyration  is  independent  of  the  thickness  of  the  lamina, 
we  may  take  the  element  of  area  instead  of  the  element  of 
mass,  and  the  total  area  of  the  lamina  instead  of  its  mass. 

198.  If  A  and  B  be  the  moments  of  inertia  of  an  infi- 
nitely thin  plate,  or  lamina,  with  respect  to  two  rectangular 
axes  OX,  OY,  lying  in  its  plane,  and  if  C  be  the  moment  of 
inertia  relative  to  OZ  drawn  perpendicular  to  the  plane,  we 

have  C=A  +  B.  (4) 

For,  we  have  in  this  case  A  =  '2y2dm,  B  =  *2,x2dm,  and 
0»X(af  +  *?)«&*. 

Again,  for  every  two  rectangular  axes  in  the  plane  of  the 
lamina,  at  any  point,  we  have 

"2,x2dm  +  72ly2dm  =  const. 

Hence,  if  one  be  a  maximum,  the  other  is  a  minimum,  and 
vice  versa. 

We  shall,  in  all  investigations  concerning  laminae,  take  C 
for  the  moment  of  inertia  relative  to  a  line  perpendicular  to 
the  lamina. 


294  Integrals  of  Inertia. 

199.    Uniform    Rod,    Rectangular    Lamina. — We 

commence  with  the  simple  case  of  a  rod,  the  axis  being  perpen- 
dicular to  its  length,  and  passing  through  either  extremity. 

Let  x  be  the  distance  of  any  element  dm  of  the  rod  from 
the  extremity ;  then,  since  the  rod  is  uniform,  dm  is  propor- 
tional to  dx,  and  we  may  assume  dm  =  fidx:  hence,  the 
moment  of  inertia  /is  represented  by  /u2a?2d#,  or  by 


. 


x*dxf 


where  /  is  the  length  of  the  rod. 

IX  I3  P 

Hence  I=  —  =  M-. 

3  3 

If  the  axis  be  drawn  through  the  middle  point  of  the  rod, 
perpendicular  to  its  length,  the  moment  of  inertia  is  plainly 
the  same  for  each  half  of  the  rod,  and  we  shall  have  in  this  case 

12 

Next,  let  us  take  a  rectangular  lamina,  and  suppose  the 
axis  drawn  through  its  centre,  parallel  to  one  of  its  sideSr 

Here,  it  is  evident  that  the  lamina  may  be  regarded  as 
made  up  of  an  infinite  number  of  parallel  rods  of  equal 
length,  perpendicular  to  the  axis,  each  having  the  same 
radius  of  gyration,  and  consequently  the  radius  of  gyration 
of  the  lamina  is  the  same  as  that  of  one  of  the  rods. 

Accordingly,  we  have,  denoting  the  lengths  of  the  sides 
of  the  rectangle  by  2a  and  2#,  and  the  moments  of  inertia 
round  axes  through  the  centre  parallel  to  the  sides,  by  A  and 
B,  respectively, 

A  =  -Mb\    B=-Ma\  (5) 

3  3 

Hence  also,  by  (4),  the  moment  of  inertia  round  an  axis 
through  the  centre  of  gravity  and  perpendicular  to  the  plane 
of  the  lamina,  is 

-  if  (a2  +  b2).  (6) 

o 
By  applying  the  principle  of  Art.  196  we  can  nOw  find 
its  moments  of  inertia  with  respect  to  any  right  line  either 
lying  in,  or  perpendicular  to,  the  plane  of  the  lamina. 


Circular  Plate,  Cylinder.  295 

200.  Rectangular  Parallelepiped. — Since  a  parallel- 
epiped may  be  conceived  as  consisting  of  an  infinite  number 
of  laminae,  each  of  which  has  the  same  radius  of  gyration 
relative  to  an  axis  drawn  perpendicular  to  their  planes,  it 
follows  that  the  radius  of  gyration  of  the  parallelepiped  is 
the  same  as  that  of  one  of  the  laminae. 

Hence,  if  the  length  of  the  sides  of  the  parallelepiped  be 
2a,  2b,  and  2c,  respectively;  and,  if  A,  B,  Obe  respectively 
the  moments  of  inertia  relative  to  three  axes  drawn  through 
the  centre  of  gravity,  parallel  to  the  edges  of  the  parallel- 
epiped, we  have,  by  the  last, 

A  =  -M(b2  +  c2),    B  =  -M(c'i  +  a2),     C  =  -M(a*+b2).        (7) 

O  O  o 

201.  Circular  Plate,  Cylinder. — If  the  axis  be 
drawn  through  its  centre,  perpendicular  to  the  plane  of  a 
circular  ring  of  infinitely  small  breadth,  since  each  point  of 
the  ring  may  be  regarded  as  at  the  same  distance  r  from  the 
axis,  its  moment  of  inertia  is  r2dm,  where  dm  represents  its 
mass. 

Hence,  considering  each  ring  as  an  element  of  a  circular 
plate,  and  observing  that  dm  =  p2wrdr9  we  get  for  C,  the 
moment  of  inertia  of  the  circular  plate  of  radius  a, 

C-2wA*f*dr-^  =  M^; 

Jo  2  2 

Consequently,  the  moment  of  inertia  of  a  ring  whose 
outer  and  inner  radii  are  a  and  b,  respectively,  with  respect  to 
the  same  axis,  is 

|»    ■:  «4  -V      „ra2  +  b2 

27TU         rdr  =  TTLL =  ML . 

J&  2  2 

Again,  by  (4),  the  moment  of  inertia  of  a  circular  plate 

a2 
about  any  diameter  is  M-,  since  the  moments  of  inertia  are 

4 
obviously  the  same  respecting  all  diameters. 

In  like  manner,  the  moment  of  inertia  of  a  ring  relative 
to  any  diameter  is 

nr  a2  +  b°' 


296 


Integrals  of  Inertia. 


Also,  the  moment  of  inertia  of  a  right  cylinder  about  its 
axis  of  figure  is 


a  being  the  radius  of  the  section  of  the  cylinder. 

Again,  the  moment  of  inertia  relative  to  any  edge  of  the 

■3 

cylinder  is  -  Ma2. 

202.  Right  Cone. — To  find  the  moment  of  inertia  of  a 
right  cone  relative  to  its  axis,  we  conceive  it  divided  into  an 
infinite  number  of  circular  plates,  whose  centres  lie  along  the 
axis ;  and,  denoting  by  x  the  distance  of  the  centre  of  any 
section  from  the  vertex  of  the  cone,  and  by  a  the  semi-angle 
of  the  cone,  we  have 


7r/utan4o  [h 


1: 


x*dx 


TTfib'h 


10 


where  h  is  the  height  of  the  cone,  and  b  the  radius  of  its  base. 
Hence,  since  by  Art.  169  the  volume  of  the  cone  is  -  b2h, 
we  have 


I=±Mb\ 
10 


(8) 


203.  Elliptic  Plate. — Next  let  us  suppose  the  lamina 
an  ellipse,  of  semi-axes  a  and  b ;  and 
let  A  and  B  be  the  moments  of  inertia 
relative  to  these  axes,  respectively. 

Describe  a  circle  with  the  axis 
minor  for  diameter,  and  suppose  the 
lamina  divided  into  rods  by  sections 
perpendicular  to  this  axis.  Let  &  be 
the  moment  of  inertia  for  the  circle 
round  its  diameter. 

Then,  denoting  by  dB  and  dl?  the  moments  of  inertia  of 
corresponding  rods,  we  have 


Fig-  47- 


dB'.dB  =  {np)' :  (np'f  =  (oa)3  :  (ob)3  =  a3 :  V 
.-.  B:B'=a3:b3. 


But  B',  by  Art.  201,  is 


Sphere.  297 

M'b2 


to     M'  «3     ilf  , 
•\  J5  =  —  —  =  —  a2. 
4    0       4 

Similarly,  -4  =  —  b2. 

Hence  the  moment  C  round  a  line  through  the  centre  of 
the  ellipse,  perpendicular  to  its  plane,  is 

*(*+»).  (9) 

It  is  plain,  as  before,  that  the  expression  for  the  moment 
of  inertia  of  an  elliptical  cylinder  relative  to  its  axis  is  of  the 
same  form. 

204.  Spbere. — If  we  suppose  a  sphere  divided  into  an 
infinite  number  of  concentric  spherical  shells,  the  moment  of 
inertia  of  each  shell  is  plainly  the  same  for  all  diameters ; 
and  accordingly,  representing  the  mass  of  any  element  of  a 
shell  by  dm,  and  by  x,  y,  z  any  point  on  it,  we  have 

*2x2dm  =  *2y2dm  =  ^%%dm. 

But  2(x*  +  y2  +  z2)dm=  2r2dm; 

2 
.*.  S (x%  +  y2) dm  =  -  *2r2dm. 
3 
Hence,  (a)  the  moment  of  inertia  of  a  shell  whose  radius 

2 
is  r  with  respect  to  any  diameter  is  -  mr2,  where  m  repre- 

o 
sents  the  mass  of  the  shell. 

Again,  (b)  for  a  solid  sphere  of  radius  B,  since  the  volume 
of  an  indefinitely  thin  shell  of  radius  r  is  $irr2dr,  we  get 


2r*eft>  =  4?r 


iAdr  =  ^TrR5  =  ^VB? 


0  5  5 

"When  this  is  substituted,  the  moment  of  inertia  of  a  solid 
homogeneous  sphere  relative  to  any  diameter  is  found  to  be 

-MM2.  (10) 

5  v     ; 


298  Integrals  of  Inertia. 

205.  Ellipsoid. — Let  the  equation  of  an  ellipsoid  be 
x2     y2     z2 

and  suppose  At  B,  C  to  be  the  moments  of  inertia  relative  to 
the  axes  a,  b,  c,  respectively ;  then 

C  =  juS  (*•  +  y2)  dV  =  /*       [  (x2  +  y2)  dxdydn. 


Now,  let 
and  we  get 


=  *,     i  =  t/ 


y    „>    *    _, 


C  m  juabc 


[l  (a?x'2+b2y'2)dx'dy'dz', 


where  the  integrals  are  extended  to  all  points  within  the 
sphere 

x2  +  i/2  +  z'2  =  1. 
But,  by  the  last  example  we  have 

\\\x'2dx'dy'dz'  =  [[[y'2dx'dy'dz'  =  —  tt; 

.*.  C  =  —  TT/iabc  (a2  +  b2)=  —  (a2  +  b2).  (11) 

In  like  manner, 

A  =  -{b2  +  c2),    B--=—(c2  +  a2). 

It  should  be  remarked  that  the  moments  of  inertia  of  the 
ellipsoid  with  respect  to  its  three  principal  planes  are 

—  a2,     —  b29     —  c2,  respectively. 

o  o  0 


Moments  of  Inertia  of  a  Lamina.  299 

206.  Moments  of  Inertia  of  a  Lamina. — Suppose 
that  any  plane  lamina  is  referred  to  two  rectangular  axes 
drawn  through  any  origin  0,  and  that  a  is  the  angle  which 
any  right  line  through  0,  lying  in  the  plane,  makes  with  the 
axis  of  x ;  then,  if  /  be  the  moment  of  inertia  of  the  lamina 
relative  to  this  line,  we  have 

i"  =  ^p2dm  =  2  (y  cos  a  -  x  sin  a)2  dm 

=  cos2a  '2>y2dm  +  sin2a y2lx2dm  -  2  sin  a  cos  a'Zxydm 

=  a  cos2a  +  b  sin2  a  -  ih  sin  a  cos  a;  (12) 

where  a  and  b  represent  the  moments  of  inertia  relative  to 

the  axes  of  x  and  y,  respectively ;  and  h  is  the  product  of 

inertia  relative  to  the  same  axes. 

Again,  supposing  X  and!Fto  be  the  co-ordinates  of  a  point 

taken  on  the  same  line  at  a  distance  R  from  the  origin,  we 

X  Y 

get  cos  a  =  -s-,  sin  a  =  -=- ;  and,  consequently, 
M  It 

IR2  =  aX2  +  bY*-2hXY. 

Accordingly,  if  an  ellipse  be  constructed  whose  equation  is 

aX2  +  bY2  -  2hXY=  const.,  (13) 

we  have 

IE2  =  const. ; 

and,  consequently,  the  moment  of  inertia  relative  to  any  line 
drawn  through  the  origin  varies  inversely  as  the  square  of 
the  corresponding  radius  vector  of  this  ellipse. 

The  form  and  position  of  this  ellipse  are  evidently  inde- 
pendent of  the  particular  axes  assumed  ;  but  its  equation  is 
more  simple  if  the  axes,  major  and  minor,  of  the  ellipse  had 
been  assumed  as  the  axes  of  co-ordinates.  Again,  since  in 
this  case  the  coefficient  of  J7  disappears  from  the  equa- 
tion of  the  curve,  we  see  that  there  exists  at  every  point  in 
a  body  one  pair  of  rectangular  axes  for  which  the  quantity 
h  or  ^xydm  =  o. 

This  pair  of  axes  is  called  the  principal  axes  at  the 
point ;  and  the  corresponding  moments  of  inertia  are  called  the 
principal  moments  of  inertia  of  the  lamina  relative  to  the  point. 


300  Integrals  of  Inertia, 

Again,  if  A  and  B  represent  the  principal  moments  of 
inertia,  equation  (12)  becomes 

1  =  A  cos2a  +  B  sin2a.  (14) 

Hence,  for  a  lamina,  the  moment  of  inertia  relative  to 
any  axis  through  a  point  can  be  found  when  the  principal 
moments  relative  to  the  point  are  determined. 

The  equation  of  the  ellipse  (13)  becomes,  when  referred 
to  the  principal  axes, 

AX2  +  BY2  =  const. 

207.  Momenta!  Ellipse. — Since  the  moments  of  inertia 
for  all  axes  are  determined  when  those  relative  to  the  centre 
of  gravity  are  known,  it  is  sufficient  to  consider  the  case 
where  the  origin  is  at  the  centre  of  gravity.  With  reference 
to  this  case,  the  ellipse 

AX2  +  BY2  =  const.  (15) 

is  called  the  momental  ellipse  of  the  lamina. 

Again,  if  two  different  distributions  of  matter  in  the 
same  plane  have  a  common  centre  of  gravity,  and  have  the 
same  principal  axes  and  principal  moments  of  inertia,  at 
that  point,  they  have  the  same  moments  of  inertia  relative  to 
all  axes. 

This  is  an  immediate  consequence  of  (14).  Hence  it  is 
easily  seen  that  the  moments  of  inertia  for  any  lamina  are 

M 

the  same  as  for  the  system  of  four  equal  masses,  each  — , 

placed  on  the  two  central  principal  axes,  at  the  four  dis- 
tances ±  a  and  ±  b,  from  the  centre  of  gravity,  where  a  and  b 
are  determined  by  the  equations 

A  =  -Mb\    B=-Ma2. 

2  2 

Again,  if  two  systems  of  the  same  total  mass,  in  a  plane, 
have  a  common  centre  of  gravity,  and  have  equal  moments 
of  inertia  relative  to  any  three  axes,  through  their  common 
centre  of  gravity,  they  have  the  same  moments  of  inertia  for 
all 


Momen  tal  Ellipse.  301 

This  follows  immediately  since  an  ellipse  is  determined 
when  its  centre  and  three  points  on  its  circumference  are 
given. 

Again,  it  may  be  observed  that  the  boundary  of  an 
elliptical  lamina  may  be  regarded  as  the  momental  ellipse  of 
the  lamina. 

For,  if  I  be  the  moment  of  inertia  relative  to  any 
diameter  making  the  angle  a  with  the  axis  major,  we  have 

7=  A  cos2a  +  B  sin2a. 
But,  by  Art.  203, 

4  4 

M 
/-  I-  —  (b2  cos2 a  +  a*  sin2 a) 

4  ' 


M  97„/cos2a      sin 
=  —  a2b2[ — —  + 
4         \   a% 

Ma2b2 
4    r2 


in2a\ 


Hence  the  moment  of  inertia  varies  inversely  as  the  square 
of  the  semi-diameter  r ;  and,  consequently,  the  ellipse  may  be 
regarded  as  its  own  momental  ellipse. 

208.  Products  of  Inertia  of  Lamina. — Suppose  the 
lamina  referred  to  its  principal  axes  at  a  point  0 ;  and  let  p 
and  q  be  the  distances  of  any  element  dm  from  two  axes, 
which  make  the  angles  a  and  [5  with  the  axis  of  x  ;  then  we 
have 

^pqdm  =  S  (y  cos  a  -  x  sin  a) {y  cos  ]3  -  x  sin  j3)  dm 
=  cos  a  cos  j3  ^y2dm  +  sin  a  sin  j3  *2x2dm 

-  sin  (a  +  (3)  ^xydm 
=  A  cos  a  cos  ]3  +  B  sin  a  sin  j3, 

since     A  =  *2y2dm,     B  =  'Sx^dm,  and  ^xydm  =  o. 
Hence,  if  'Spqdm  =  o,  we  have 

A  cos  a  cos  /3  +  B  sin  0  sin/3  =  o, 


302 


Integrals  of  Inertia. 


and  accordingly  the  axes  are  a  pair  of  conjugate  diameters 
of  the  momental  ellipse 

AX2  +  BY2  =  const. 


Hence,  if  two  laminae  in  the  same  plane  have  for  any  point 

two  pairs  of  axes  for  which  *2pqdm  =  o  and  "Ep^dm  =  o, 

they  have  the  same  principal  axes  at  the  point.     This  follows 

from  the  easily  established  property,  that  if  two  ellipses  have 

two  pairs  of  conjugate  diameters  in  common,  they  must  be 

similar  and  coaxal. 

209.  Triangular   Lamina   and   Prism. — Suppose    a 

triangular  lamina,  whose  sides  are  a,  b,  c,  to  be  divided  into 

a  system  of  rods  parallel  to  a  side  a  ; 

and  let  A  represent  the  moment  of 

inertia  relative  to  a  line  parallel  to 

the  side  a,  and  drawn  through  the 

opposite  vertex;    also  let  p  be  the 

perpendicular    of    the    triangle    on 

the  side  a,  and  x  the  distance  of  an 

elementary  rod  from  the  vertex;  then  Fi      g 

we  have,  since  the  mass  dm  of  the 

ax 
elementary  rod  may  be  represented  by  /m — dx, 


ax 


A  =  ^x2dm  =  u^x^—dx 
P 


a  [*  m  ,         ap% 
=  u  -     xzdx  =  u—  =  — f 
FJo  4 


M 

2 


In  like  manner,  let  B  and  C  be  the  moments  of  inertia 
relative  to  lines  drawn  through  the  other  vertices  parallel  to 
b  and  c ;  and  let  q,  r  be  the  corresponding  perpendiculars  of 
the  triangle,  and  we  have 


»-?* 


2 


Again,  if  A0,  B0,  C0,  represent  the  moments  of  inertia 


Triangular  Lamina  and  Prism.  303 

relative  to  three  parallels  to  the  sides,  drawn  through  the 
centre  of  gravity  of  the  lamina,  we  have,  by  (2), 

AQ  =  -Mp\     B0=\Mq\      C0  =  ±Mr\        (16) 

Io  Io  10 

Also,  if  Ai,  2?i,  Cl9  be  the  moments  of  inertia  relative 
to  the  sides  a,  b,  c,  respectively,  it  follows,  in  Eke  manner, 
from  (2),  that 

Ax=1tMP\     Bx  =  \Mq\      Cx-\Mr\  (17) 

000 

Again,  it  is  readily  seen  that  the  values  of  A,  A0,  Ax,  &c, 
are  the  same  as  if  the  whole  mass  If  were  divided  into  three 
equal  masses,  placed  respectively  at  the  middle  points  of  the 
sides  of  the  lamina. 

Consequently,  by  Art.  207,  the  moments  of  inertia  of  the 
triangular  lamina  relative  to  all  axes  are  the  same  as  for 

three  masses,  each  — ,  placed  at  the  middle   points  of  the 

o 
sides  of  the  triangle. 

Hence,  if  i"  be  the  moment  of  inertia  of  a  triangular 
lamina  with  respect  to  the  perpendicular  to  its  plane  drawn 
through  its  centre  of  gravity,  we  have 

I=^-M{az  +  62  +  c2).  (18) 

This  expression  also  holds  for  the  moment  of  inertia  of  a 
right  triangular  prism  with  respect  to  its  axis* 

In  like  manner  the  moments  of  inertia  of  the  triangular 
lamina  relative  to  the  three  perpendiculars  to  its  plane, 
drawn  through  its  vertices,  are 


'A 


-\    -M(ci  +  az--\    -A 


3/      4      \  37     4 

and  the  same  expressions  hold  for  a  triangular  prism  relative 
to  its  edges. 

*  By  the  axis  of  a  prism  is  understood  the  right  line  drawn  through  its 
centre  of  gravity  parallel  to  its  edges. 


304  Integrals  of  Inertia. 

210.  Momental  Ellipse  of  a  Triangle. — It  can   be 

shown  without  difficulty  that  the  ellipse  which  touches  at  the 
middle  points  of  the  sides 
may  be  taken  for  the  mo- 
mental  ellipse  of  the  triangle. 
For,  let  x,  y,  z  be  the 
middle  points  of  the  sides, 
and  it  is  easily  seen  that  o 
is  the  centre  of  this  ellipse ; 
also,  if  Ji,  72,  I9  be  the 
moments  of    inertia    of  the  lg* 49' 

lamina  relative  to  the  lines  ax,  by,  cz,  respectively,  it  can  be 
readily  shown  from  (17),  that  we  have 


J» :  It :  JT,  - 


(axf  '  (by)*  '  («)• 


(oxy  (oyy  (ozy 

Accordingly,  by  Art.  207,  the  ellipse  xyz  may  be  taken  for 
the  momental  ellipse  of  the  lamina. 

211.  Tetrahedron. — If  a  solid  tetrahedron  be  supposed 
divided  into  thin  laminae  parallel  to  one  of  its  faces,  and  if 
A,  B,  C,  D  represent  its  moments  of  inertia  with  regard 
to  the  four  planes  drawn  respectively  through  its  vertices 
parallel  to  its  faces ;  then,  denoting  the  areas  of  the  corre- 
sponding faces  by  a,  b,  c,  d,  and  the  corresponding  perpen- 
diculars of  the  tetrahedron  by  p,  q,  r,  s,  respectively,  it  is 
easily  seen,  as  in  Art.  209,  that  we  shall  have 

x2  a  [p 

A  =  lux%dm  =  a^x^a—  dx  =  u—\   x*dx 
f  VJo 

ap%      3 

In  like  manner  we  have 

B  =  ^-Mq\     C^Mr*,     D=±Ms\ 
5  o  0 


Solid  Ring. 


305 


Again,  if  A0i  B0,  C0,  D0  be  the  corresponding  moments  of 
inertia  relative  to  the  parallel  planes  drawn  through  the 
centre  of  gravity  of  the  tetrahedron,  we  have,  by  (2), 

Ao  =  loMp\    B.-^Mf,     ft-^-K*.    D«=±Ms\   (19) 

Also,  if  Ai,  JBi,  Ch  Dx  be  the  moments  of  inertia  relative 
to  the  four  faces  of  the  tetrahedron,  we  have 

Ax  =  —  Mp\    B,  =  —  Mq%     d  =  —  Mr\    A  =  —  Ms\  (20) 
10  10  10  10  v     ' 

212.  Solid  Ring.* — If  a  plane  closed  curve,  which  is 
symmetrical  with  respect  to  an  axis  AB}  be  made  to  revolve 
round  a  parallel  axis,  lying  in 
its  plane,  but  not  intersecting  the 
curve,  to  prove  that  the  moment 
of  inertia  i"  of  the  generated  solid, 
taken  with  respect  to  the  axis  of 
revolution,  is  represented  by 


Fig.  50. 


where  M  is  the  mass  of  the  solid,    q 

h  the  distance  between  the  parallel 
axes,  and  k  the  radius  of  gyration 
of  the  generating  area  relative  to  its  axis. 

For,  if  the  axis  of  revolution  be  taken  as  the  axis  of  xt 
and,  if  y,  Y  be  the  distances  of  any  point  P  within  the 
generating  area  from  AB,  and  from  OX,  respectively ;  and, 
if  dA  be  the  corresponding  element  of  the  area,  then  the 
volume  of  the  elementary  ring  generated  by  dA  is  2k  Yd  A, 
and  its  mass  2irix  Yd  A  ;  hence  the  moment  of  inertia  of  this 
elementary  ring,  relative  to  the  axis  of  X,  is  2wfjiY3dA. 
Accordingly,  we  have 

2=  27rfjL^YzdA  =  27Tfi^(h  +  yYdA 

=  27T/JS  (h*  +  $h*y  +  shy^  +  y3)  dA. 


*  The  theorems  of  this  Article  were  given  by  Professor  Townsend  in  the 
Quarterly  Journal  of  Mathematics ,  1869. 


[30] 


306  Integrals  of  Inertia. 

Moreover,  since  the  curve  is  symmetrical  with  respect  to 
the  axis  AB,  it  is  easily  seen  that  we  have 

2ydA  =  o,     SyHA  =  o. 

Also,  by  definition,     2y2dA  =  AW". 

Hence  I  =  2tt/u  hA  (h2  +  3A2). 

Again,  by  Art.  177,     M '=  nrfihA  ; 

.-.  I=M{h2  +  sk2).  (21) 

This  leads  immediately  to  some  important  cases. 

Thus,  for  example,  the  moment  of  inertia  of  a  circular 
ring,  of  radius  a,  round  its  axis  is 


Mfe+tA. 


Again,  if  a  square  of  side  a  revolve  round  any  line  in  its 
plane,  situated  at  the  distance  h  from  its  centre,  we  have 

I  =  M{h2  +  a2). 

There  is  no  difficulty  in  adding  other  examples. 

213.  General  Expression  for  Products  of  Inertia. 

— "We  shall  conclude  this  Chapter  with  a  short  discussion  of 
the  general  case  of  the  moments  and  products  of  inertia,  for 
any  body,  or  system. 

Let  us  suppose  the  system  referred  to  three  rectangular 
planes,  and  let  p,  q,  r  represent  the  respective  distances  of 
any  element  dm  from  the  three  planes 

x  cos  a  +  y  cos  j3  +  z  cos  7  =  0, 

x  cos  a  +  y  cos  j3'  +  z  cos  y  =  o, 

x  cos  a"  +  y  cos  j3"  +  s  cos  7"  =  o. 
Then 

'2lpqdm=^l(xco8a+ycosP+zco8y)(xcosa+ycosf5/+zco8y/)dm 

«=  cos  a  cos  a  1.x2  dm  +  cos  fi  cos  (3' ?,y2  dm  +  cosy  cosy' Sz2  dm 

+  (cos  a  cos  j3'  +  cos /3  cos  a)  Sxydm 

+  (cos  7  cos  a  +  cos  a  cos  7')  Szxdm 

+  (cos  /3  cos  y  +  cos  7  cos  j3')  Syzdm; 

and  we  get  similar  expressions  for  Iprdm  and  Sqrdm. 


Principal  Axes.  307 

Now,  suppose  that  we  take 

'2x2dm  =  a,     ^y~dm  =  b,     Ss2c?m  =  c, 
"Syzdm  =/,     *2>xzdm  =  g,     *2xydm  =  h ; 
then  the  preceding  equation  may  be  written 

Ypqdm  =  cos  a  (a  cos  a  +  h  cos  j3'  +  g  cos  7') 
+  cos  j3  (h  cos  a  +  b  cos  j3'  +  /  cos  7') 
+  cos  7  (g  cos  a  +  f  cos  j3'  +  c  cos  7')  ;  (22) 

along  with  similar  expressions  for  'Srpdm  and  'Sqrdrn. 

214.  Principal  Axes. — Next,  let  us  suppose  that  the 
planes  are  so  assumed  as  to  satisfy  the  equations 

'Epqdm  =  o,     "2rpdm  =  o,     ^qrdm  =  o ; 

then  it  is  easily  seen*  that  these  planes  are  a  system  of  con- 
jugate diametral  planes  in  the  ellipsoid  represented  by  the 
equation 

aX2  +  bY2  +  cZ2  +  2/YZ  +  2gZX  +  2hXY  =  const.     (23) 

Hence  it  follows  that  at  any  point  there  exists  one  system  of 
rectangular  planes  for  which  the  corresponding  products  of 
inertia,  for  any  body,  vanish :  viz.,  the  principal  planes  of  the 
preceding  ellipsoid.^ 

These  three  planes  are  called  the  principal  planes  of  the 
body  relative  to  the  point,  and  the  right  lines  in  which  they 
intersect  are  called  the  principal  axes  for  the  point. 

Again,  every  two  solids  have  for  every  point  at  least  one 
common  system  of  planes  for  which  *2pqdm  =  o,  *2rpdm  =  o, 
*2qrdm  =  o,  ^pq'dm  =  o,  ILrpdrri  =  o,  ^q'rdm'  =  o; 
where  the  unaccented  letters  refer  to  the  elements  of  one 
solid,  and  the  accented  to  those  of  the  other. 

This  is  obvious  from  the  property  that  every  two  con- 
centric ellipsoids  have  one  common  system  of  diametral  planes. 


*.   Salmon's  Geometry  of  Three  Dimensions,  Art.  72. 

t  The  exceptional  cases  when  the  ellipsoid  is  of  revolution,  or  is  a  sphere, 
will  he  considered  subsequently. 

[20  a] 


308  In  tegrah  of  In  ertia . 

Again,  if  two  solids  have  for  any  point  more  than  one 
system  of  planes  for  which  the  foregoing  six  products  of 
inertia  vanish,  they  must  have  the  same  principal  planes  at 
the  point.  This  follows  since  the  two  ellipsoids  in  that  case 
must  be  similar  and  coaxal. 

215.  Principal  Moments  of  Inertia. — Let  us  now 
suppose  the  co-ordinate  planes  to  be  the  principal  planes  of 
the  body  for  the  origin,  then  the  moment  of  inertia  relative 
to  the  plane 

x  cos  a  +  y  cos  j3  +  f  cos  7  =  o 
is 

*2p2dm  =  2  (x  cos  a  +  y  cos  |3  +  I  cos  y)2dm 

=  cos2a2#2c?m  +  cos2j3  *2y2dm  +  cos2  7  ^z2dm,     (24) 

since  in  this  case  we  have 

"2xydm  =  o,     'Szxdm  =  o,     "2yzdm  =  o. 

Again,  let  /  be  the  moment  of  inertia  of  the  body  relative 
to  the  line  through  the  origin  whose  direction  angles  are 
a,  j3,  7  ;  then  we  have 

/+  2p2dm  =  2r2dm  =  2(^  +  y2  +  z2)dm; 

,\  I  =  cos2  a  S  (y2  +  z2)  dm  +  cos2j3  S  (z2  +  x2)  dm 

+  cos2  7  S  (x2  +  y2)  dm ; 

or  I  =  A  cos2a  +  B  cos2j3  f  C  cos27,  (25) 

where  A,  JB,  C  are  the  moments  of  inertia  of  the  body 
relative  to  its  three  principal  axes. 

A,  B,  C  are  called  the  three  principal  moments  of  inertia 
of  the  body  relative  to  the  origin. 

If  the  centre  of  gravity  be  taken  as  the  origin,  the 
corresponding  values  of  A,  B,  C  are  called  the  principal 
moments  of  inertia  of  the  body. 

We  suppose,  in  general,  that  A  is  the  greatest,  and  C  the 
least  of  the  three  principal  moments. 

It  follows  from  (25)  that  the  moment  of  inertia  of  a  body 
relative  to  any  line  passing  through  a  given  point  is  known, 
whenever  the  angles  which  the  line  makes  with  the  principal 
axes  are  known,  as  also  the  moments  of  inertia  relative  to 
these  axes. 


Momental  Ellipsoid.  309 

216.  Ellipsoid  of  Gyration. — Suppose,  as  before,  the 
solid  referred  to  its  three  principal  axes  at  any  point,  and  let 
a,  b,  c  be  the  corresponding  radii  of  gyration,  i.e.  let 

A  =  Ma2,     B  =  Mb2,     0  =  Mc% 

and  /=  Mk2;  then  equation  (25)  becomes 

k2  =  a2  cos2  a  +  b2  cos2/3  +  c2cos2y.  (26) 

Now,  if  we  suppose  an  ellipsoid  described  having  the 
principal  axes  for  the  directions,  and  a,  b,  c  for  the  lengths 
of  its  corresponding  semi-axes  ;  then  (26)  shows  that  the 
radius  of  gyration  of  the  body,  relative  to  the  perpendicular 
from  the  origin  on  any  tangent  plane  to  this  ellipsoid,  is 
equal  in  length  to  this  perpendicular.  (Salmon's  Geometry 
of  Three  Dimensions,  Art.  89.) 

The  foregoing  ellipsoid  is  called  the  ellipsoid  of  gyration 
relative  to  the  point.  It  should,  however,  be  observed  that 
by  the  ellipsoid  of  gyration  of  a  body  is  meant  the  ellipsoid 
in  the  particular  case  where  the  origin  is  at  the  centre  of 
gravity  of  the  body. 

217.  Honiental  Ellipsoid. — If  X,  Y,  Z  be  the  co- 
ordinates of  a  point  R  taken  on  the  right  line  through  the 
origin  0,  whose  direction  angles  are  a,  /3,  7,  we  have 

X=  Oleosa,     Y=OB  cosj3,    Z  =  OR  cosy. 

Substituting  the  values  of  cos  a,  cos  j3,  cos  y,  deduced 
from  these  equations,  in  (25),  it  becomes 

/.  OR2  =  AX2  +  BY2  +  CZ2. 

Suppose,  now,  that  the  point  R  lies  on  the  ellipsoid 

AX2  +  BY2+  CZ2  =  const.,  (27) 

and  we  get  I .  OR2  =  A,  denoting  the  constant  by  A  ; 

■■■T=m-  w 

Hence  the  moment  of  inertia  relative  to  any  axis,  drawn 
through  the  origin,  varies  inversely  as  the  square  of  the  cor- 
responding diameter  of  the  ellipsoid  (27). 


310  Integrals  of  Inertia. 

J 

From  this  property  the  ellipsoid  is  called  the  momenta! 
ellipsoid  at  the  point. 

When  the  origin  is  taken  at  the  centre  of  gravity  of  the 
body,  this  ellipsoid  is  called  the  central  ellipsoid  of  the  body. 

If  two  of  the  principal  moments  of  inertia  relative  to  any 
point  be  equal,  the  momental  ellipsoid  becomes  one  of  re- 
volution, and  in  this  case  all  diameters  perpendicular  to  its 
axis  of  revolution  are  principal  axes  relative  to  the  point. 

If  the  three  principal  moments  at  any  point  be  equal,  the 
ellipsoid  becomes  a  sphere,  and  the  moments  of  inertia  for  all 
axes  drawn  through  the  point  are  equal.  Every  such  axis  is 
a  principal  axiB  at  the  point. 

For  example,  it  is  plain  that  the  three  principal  moments 
for  the  centre  of  a  cube  are  equal,  and,  consequently,  its 
moments  of  inertia  for  all  axes,  through  its  centre,  are  equal. 

218.  Equimomental  Cone. — Again,  since 

COS8  a  +  COS2j3  +  COS2  7  =  I, 

equation  (25)  may  be  written  in  the  form 

(A  -  /)  cos2a  +  (B-I)  cos2/3  +  (G-  I)  cos27  =  o  ; 

hence  the  equation 

(A  -  I)  X*  +  (B  -I)Y*+(C-I)Z*  =  o         (29) 

represents  a  cone  such  that  the  moment  of  inertia  is  the  same 
for  each  of  its  edges.  Such  a  cone  is  called  an  equimomental 
cone  of  the  body. 

Again,  the  three  axes  of  any  equimomental  cone,  for  any 
solid,  are  the  principal  axes  of  the  solid  relative  to  the  vertex 
of  the  cone. 

When  1=  By  the  cone  breaks  up  into  two  planes ;  viz., 
the  cyclic  sections  of  the  momental  ellipsoid. 

For  a  more  complete  discussion  of  the  general  theory  of 
moments  of  inertia  and  principal  axes,  the  student  is  referred 
to  Routh's  Rigid  Dynamics,  chapters  1.  and  11. ;  as  also  to 
Professor  Townsend's  papers  in  the  Camb.  and  Dub.  Math. 
Journal,  1846,  1847. 


Examples.  311 


Examples. 


Find  the  expressions  for  the  moments  of  inertia  in  the  following,  the  bodies 
being  supposed  homogeneous  in  all  cases  : — 

i.  A  parallelogram,  of  sides  a,  b,  and  angle  0,  with  respect  to  its  sides. 

Ans.  —  b2  sin2  0,     —  a2  sin2  9. 
3  3 

2.  A  rod,  of  length  o,  with  respect  to  an  axis  perpendicular  to  the  rod  and 
at  a  distance  d  from  its  middle  point. 

Ans.  M  l-  +  d2\. 

3.  An  equilateral  triangle,  of  side  a,  relative  to  a  line  in  its  plane  at  the 
distance  d  from  its  centre  of  gravity. 

Ans. 


L  *(-**) 


4.  A  right-angled  triangle,  of  hypothenuse  c,  relative  to  a  perpendicular  to 
its  plane  passing  through  the  right  angle. 

c2 


Ans.  If- 
6 


5.  A  hollow  circular  cylinder,  relative  to  its  axis. 


r2  +  r'2 
Ans.  M ,  where  r  and  r'  are  the  radii  of  the  bounding  circles. 

6.  A  truncated  cone  with  reference  to  its  axis. 

3  M  b5  -  b'5 
Ans.  —  r= — —,  where  b  and  b'  are  the  radii  of  its  bases. 
10  b3  -b* 

7.  A  right  cone  with  respect  to  an  axis  drawn  through  its  vertex  perpen- 
dicular to  its  axis. 

7M  /         b2\ 
Ans.  —  [h2  +  —  ) ,  where  h  denotes  the  altitude  of  the  cone, 
5    \         4/ 
and  b  the  radius  of  its  base. 

8.  An  ellipsoid  with  respect  to  a  diameter  making  angles  a,  £,  y  with  its 


Ans.  —  [a2  sin2a  +  b2  sin2/3  +  c2  sin27  J . 


9.  Area  bounded  by  two  rectangles  having  a  common  centre,  and  whose 
sides  are  respectively  parallel,  with  respect  to  an  axis  through  their  centre 
perpendicular  to  the  plane. 

.        M(a2  +  b2)ab-(a'2+b'Aa'b' 

Ans.  —~ ,        ... ' . 

12  ab  -  a'b' 


312  Examples, 

io.  A  square,  of  side  a,  relative  to  any  line  in  its  plane,  passing  through  its 
centre. 


fl2 

Ans.  M-. 
12 


II.  A  regular  polygon,  or  prism,  with  respect  to  its  axis. 


Ans.  —  ( -R2  +  2r2  J ,  where  R  and  r  are  the  radii  of  the 

circles  circumscrihed,  and  inscribed  to  the  polygon. 

u.  Prove  that  a  parallelogram  and  its  maximum  inscribed  ellipse  have  the 
same  principal  axes  at  their  common  centre  of  figure. 

13.  Prove  that  the  moments  and  products  of  inertia  of  any  triangular 

M 
lamina,  of  mass  M,  are  the  same  as  for  three  masses,  each  — ,  placed  at  the 

three  vertices  of  the  triangle,  combined  with  a  mass  -  M  placed  at  its  centre  of 

4 
gravity. 

14.  Prove  that  the  moments  and  products  of  inertia  of  any  tetrahedron  are 

M 
the  same  as  for  four  masses,  each  — ,  placed  at  the  vertices  of  the  tetrahedron, 

4  2° 

combined  with  a  mass  -  M  placed  at  its  centre  of  gravity. 

15.  If  a  system  of  equimomental  axes,  for  any  solid,  all  lie  in  a  principal 
plane  passing  through  its  centre  of  gravity,  prove  that  they  envelop  a  conic, 
having  that  point  for  centre,  and  the  principal  axes  in  the  plane  for  axes. 

1 6.  Prove  also  that  the  ellipses  obtained  by  varying  the  magnitude  of  the 
moment  of  inertia  form  a  confocal  system. 

17.  Prove  that  the  sum  of  the  moments  of  inertia  of  a  body  relative  to  any 
three  rectangular  axes  drawn  through  the  same  point  is  constant. 

18.  Prove  that  a  principal  axis  belonging  to  the  centre  of  gravity  of  a  body 
is  also  a  principal  axis  with  respect  to  every  point  on  its  length. 

19.  Prove  that  the  envelope  of  a  plane  for  which  the  moment  of  inertia  of 
a  body  is  constant  is  an  ellipsoid,  confocal  with  the  ellipsoid  of  gyration  of  the 
body. 

20.  If  a  system  of  equimomental  planes  pass  through  a  point,  prove  that 
they  envelop  a  cone  of  the  second  degree. 

2 1 .  For  different  values  of  the  constant  moment  the  several  enveloped  cones 
are  confocal  ? 

22.  The  common  axes  of  this  system  of  cones  are  the  three  principal  axes  of 
the  body  for  the  point  ? 

23.  The  three  principal  axes  at  any  point  are  the  normals  to  the  three  sur- 
faces confocal  to  the  ellipsoid  of  gyration,  which  pass  through  the  point. 
(M.  Binet,  Jour,  de  VEc.  Foly.  1813.) 


.      (    313    ) 
CHAPTER  XI. 

MULTIPLE    INTEGRALS. 

219.  Double  Integration. — In  the  preceding  Chapters  we 
have  considered  several  cases  of  double  and  triple  integra- 
tion in  the  determination  of  volumes  and  other  problems 
connected  with  surfaces.  We  now  proceed  to  a  short  treat- 
ment of  the  general  problem  of  Multiple  Integration,  com- 
mencing with  double  integrals. 

The  general  form  of  a  double  integral  may  be  written 


J  x0Jy 


f{x,  y)dxdy, 

in  which  we  suppose  the  integration  first  taken  with  respect 
to  y,  regarding  x  as  constant.  In  this  case,  F,  y0i  the  limits 
of  y,  are,  in  general,  functions  of  x ;  and  the  limits  of  x  are 
constants. 

Let  us  take  for  example  the  integral 


r.tf. 


xl~lym~ldxdyy 
in  which  /  is  supposed  greater  than  m. 


Here 


therefore       U=  — 
m 


1  fa2m        \ 
faZm        \ 


x1-1  [z—-xm\dx  = 


P-nf 

In  many  cases  the  variables  are  to  be  taken  so  as  to  in- 
clude all  values  limited  by  a  certain  condition,  which  can  be 
expressed  by  an  inequality :  for  instance,  to  find 


-if 


x^y™'1  dxdy, 

extended  to  all  positive  values  of  x  and  y  subject  to  the  con- 
dition   x  +  y  <  h. 

[21] 


314 


Multiple  Integrals, 


Here  the  limits  for  y  are  o  and  h  -  x ;  and  the  subsequent 
limits  of  x  are  o  and  h. 


Hence 


Let  x  =  hu,  then 


nh-x 
xl-^ym~xdxdy 
0 

=  -  f  xl-Uh-x)mdx. 


w  J0      v       '  r(/  +  w+ 1) '       v  ' 

by  Art.  121. 

220.  Change  of  Order  of  Integration. — We  have 
seen  (Art.  115)  that  when  the  limits  of  x  and  y  are  con- 
stants in  a  double  integral  we  may  change  the  order  of  inte- 
gration, the  limits  remaining  unaltered.  But  when  the 
limits  of  y  are  functions  of  x,  if  the  order  of  integration  be 
changed,  it  is  necessary  to  find  the  new  limits  for  x  as  func- 
tions of  y.  This  is  usually  best  obtained  from  geometrical 
considerations. 

For  example,  in  the  integral 


U^a^f{x,y)dxdy, 


the  limits  for  y  are  given  by  the  right  line  y  =  x  and  the 
hyperbola  xy  =  a2 ;  and  the  integral 
extends  to  all  points  in  the  space 
included  by  the  hyperbola  AL>  the 
right  line  OA,  where  A  is  the  ver- 
tex of  the  hyperbola,  and  the  axis 
of  y.  Draw  AB  perpendicular  to 
the  axis  of  y.  Now  when  the  order 
of  integration  is  changed,  we  sup- 
pose the  lines  which  divide  the  area 
into  strips  taken  parallel  to  the  axis 
of  x  instead  of  the  axis  of  y.  Thus 
the  integral  breaks  up  into  two  parts — one  corresponding  to 


Change  of  Order  of  Integration,  315 

the  triangle  OAB,  the  other  to  the  remaining  area :  hence 


U= 


f(x*y)dydx  + 


fix,  y)dydx. 


As  another  example,  let  us  interchange  the  variables  in 
the  integral 


U= 


«*'  Vdxdy. 
nx 


Here,  let  OC  and  OD  be  the 

lines  represented  by  y  =  Ix   and 
y  =  mx  ;  and  let  OA  =  a.  f 

Then  the  integral  is  extended  to 
all  points  within  the  triangle  OCD. 

Accordingly,  changing  the  order,    ° 
we  get 


J 'la    ra  cm  a   rZ 

Vdydx+\  Vdydx. 

ma  Jy  J0      Jy 


A     X 


Examples. 

i.  Find  the  value  of  the  double  integral 

v_\a[*   f(y)dxdv 

h  Jo  V(a  -  x)  (x  -  y) 
Here,  changing  the  order,  the  integral  becomes 

fB  f«     f'(y)dydx 
h  Jy  V(a  —  x)(x  —  y) 

J  a  J* 

-  =  ir:  hence  Umw{f(a)  -/(0)J 
v  V(a-  x){x-y) 


2.  ProTethat 

r2a  fV2a*-*a 


"a   •a  +  V*2"!'2 

'o  K 
[21a] 


rza  ryiax-x*  ra  .•a  +  yaa-y2 

f{x,y)dxdy  =  f(x,  y),hjdx. 

JO     JO  Jo  •'«-v'a2-y9 


'M6  .  Multiple  Integral*. 

j.  Iff  pee  find  the  value  of 

j.2«  |.V^.xa  ^'(y)(^  +  y^xdxdy 
Jo    Jo  *&+-[*  +  $$' 

Ans.  ira2  (<J>(ff)  -  <f>(o, }. 

4.  Change  the  order  of  integration  in  the  double  integral 

[■2a  rylax 

U=\  Vdxdy. 

Jo    iyfuZ* 

The  limits  of  y  are  represented  by  the  circle  x2  +  y2  =  2ax,  and  the  paraboh 
y1  =  zax ;  and  we  readily  find  that 

y  Vdydx  4  „ Vdydx+\  Vdydx. 

0  Jy3  .0  JaWa'.yz  j„    }„i 


221.  Dirlchlet's  Theorem.  —  The  result  given  in 
equation  (1)  has  been  generalized  by  Dirichlet  (Liouville's 
Journal,  1839),  and  extended  to  a  large  class  of  multiple 
integrals,  as  follows  :  Wer*  j 

Commencing  with  three  variables,  let  us  consider  the  inte- 
gral 


-UJ 


xl-lym-lzn-ldxdydz, 


in  which  the  variables  are  supposed  always  positive,  and 
limited  by  the  condition 

x  +  y  +  z  <  1. 

In  this  case  the  limits  of  z  are  o  and  1  -  x  -  y  ;  those  of 
y  are  o  and  1  -  x ;  and  those  of  x,  o  and  1 . 

r  1  ri-z  ri-z-y 

Hence          U  =  x  _1  ym~l  zn~l  dx  dy  dz. 

Jo  Jo     Jo 

It  is  easily  seen,  from  (1),  that 

n-x  p.x-y  T(m)  T(n) 

Jo  Jo  r(*»  +  »  +  i) 


Dinchletfis  Theorem.  317 

therefore 

r(m)r(n)  p 

r{m  +  n+i)}0        v  ' 

r(m)  r(»)    r(flr(«  +  n+i)     r(/)  r(m)r>; 


r(m  +  n+i)'   T(l+m  +  n+  i)        r(l  +  m  +  n  +  i 


\-  w 


Again,  in  the  same  multiple  integral,  if  x,  y,  g,  being 
still  always  positive,  are  subject  to  the  condition 

x  +  y  +  s  <  h, 
we  get 

rr_,^r(0r(m)rw 

r(/+m  +  n+i)'  V^ 

This  readily  appears  by  substituting  #  =  hx\  y  =  %', 
z  =  As',  in  the  multiple  integral. 

There  is  no  difficulty  in  extending  these  results  to  any 
number  of  variables.  We  proceed  from  (3)  to  the  case  of 
four  variables ;  and  so  by  induction  to  any  number. 

Thus,  the  value  of  the  multiple  integral 

U  =  HI . .  .  x1-1  ym~J  zn~'  . . .  dxdydz 

extended  to  all  positive  values  of  x,  y,  z,  &c,  subject  to  the 
condition 

x  +  y  +  z  +  &c.  <  1 , 

lT    r(i)r(m)r(n)... 

T(i  +  l  +  m  +  n  +  ..  .)'  w 

Again,  in  the  integral 

U  *  !!!  *M  Vm~l  sM-1  dx  dy  dz,      ' 

suppose  the  variables  to  be  still  always  positive,  but  limited 
by  the  condition 


318  Multiple  Integrals. 

then  making 

3-  (?)'=•■  ©' 


par      -  /       I      m     n> 

r(  i  +-  +  -  +  - 


the  integral  transforms  into 

albmcn  C([   --i    --i     --i 

27= up     v9     wr     dudvdw, 

pqr  JJJ 

where     >.t  +  v  +u>  <  i . 
Accordingly 

v  a'i-C"r(p)ru)r0 

Again,  from  (3),  the  value  of  the  triple  integral 
jjjx^y^z^dxdydz, 

extended  to  all  positive  values,  subject  to  the  condition 

x  +  y  +  z  >  u  and  <  u  +  du9 
is 

r(or(m)r(n),.        .  .„...       r(/)r(w)r(»)  ,  tB1. 

r{i+l+m  +  n)K  r{l+m  +  n) 

Hence  the  multiple  integral 

JJT-FXtf  +  y  +  z)  xhxym-x  z""1  dxdydz, 
taken  between  the  same  limits,  has  for  its  value 

r(/)r(OT)r>) 


Dirichlefs  Theorem.  319 

Accordingly,  the  value  of  the  multiple  integral 

HI  F{x  +  y  +  z)  xl~x  ym~x  zn~x  dxdydz, 

extended  to  all  positive  values  of  the  variables,  subject  to  the 
oondition 

x  +  y  +  z  <  h, 

rgrwrwp 

r(l+m  +  n)    Jo 
In  like  manner  it  is  seen  that  if  the  multiple  integral 

JJ=  [  M(-T+  (?Y+  (*Y|  xl-'ym-xzn-ldxdydz 
be  extended  to  all  positive  values,  subject  to  the  condition 

we  have 


.  <ArM££& 


pqr        r^  +  2  +  5 

p      q      r 


rh  US  A*! 

F(u)uP  q   '     du.  (7) 


These  results  can  be  readily  extended  to  any  number  of 
variables. 


Examples. 
1.  Find  the  value  of 

extended  to  all  positive  values,  subject  to  x  +  y  <  h. 

Ans.    - — —  (eh  - 1). 

sin  In- x  ' 


820  Multiple  Integrals. 

2.  More'generally,  prove  that 

\\F'  (*+*)**■  r*dxdy  =  -At-  {F(h)  -  F(o)}, 

J  J  8111  IfT 

where  x  +  y  <  h. 

3.  Find  the  value  of 

f  J  J" .  .  dx\  dx%  .  .  .  dx„, 

extended  to  all  positive  values  of  the  variables,  subject  to  the  condition 

xx2  +  *22  +  .  •  .  +  xn2  <  &■ 

'R\n       zs* 


-  (!) 


(-3 

4.  Prove  that 


dx  dy  dz  vt 


the  integral  being  extended  to  all  positive  values  of  the  variables  for  which  the 
expression  is  real. 

5.  Show  in  general  that 

n+_l 

e  i'  r  dx\  dx-i  dx3  .  .  dxn  tt  2 

J  J  J  "  Vi"-*ia-«2*-.  .-Xn2  In  +-  i\' 

under  the  same  condition  as  in  (3). 

222.  Transformation  of  multiple  Integrals. — We 

proceed  to  consider  the  transformation  of  a  multiple  integral 
to  a  new  system  of  independent  variables. 

Suppose  it  be  required  to  transform  the  integral 

SH/fa  1/>  z)  dxdydz 

to  another  system  of  variables,  w,  v,  w,  being  given  %,  yy  z  in 
terms  of  w,  v,  to. 

This  transformation  implies  in  general  three  parts — 
(1)  the  expression  of  f  (%,  y,  z)  in  terms  of  u,  v,  w ;  (2)  the 
determination  of  the  new  system  or  systems  of  limits  ; 
(3)  the  substitution  for  dx  dy  dz. 

The  solution  of  the  first  two  questions  is  a  purely  algebraical 


Transformation  of  Multiple  Integrals.  321 

problem.     We  here  limit  ourselves  to  the  consideration  of  the 
third  question,  and  write  the  integral  in  the  form 

jdxjdySf(x,y,z)dz. 

In  the  integration  with  respect  to  z,  x  and  y  are  regarded 
as  constant ;  accordingly,  in  order  to  replace  z  by  the  new 
variable  w,  we  suppose  z  expressed,  by  means  of  the  given 
equations,  in  terms  of  x,  y,  w;  and  then  we  replace  dz  by 

—  dw.     Again,  to  transform  the  integration  from  y  to  v,  we 

must  suppose  y  expressed  in  terms  of  vt  w,  x,  and  then  dy 

replaced  by  ~  dv  :  we  next  suppose  x  replaced  by  —  du  ;  and 

we  finally  replace 

,     ,     ,     .       dz    dy  dx  _     _    _ 
dx  du  dz    by   —   —  -7-  da  dv  dw. 
J  J    dw  dv   du 

It  should  be  observed  that  in  each  of  the  latter  transfor- 
mations a  change  in  the  order  of  integration  is  supposed. 
By  this  means  the  transformed  expression  is 

,  .    dz  dy  dx    _     7     7  ._x 

(u,  v,  w)  - — —  -=-  du  dv  aw.  (8) 

^  '  dw  dv  du  ' 

where  0  (u,  v,  w)  is  the  transformation  of/  (x,  y,  z). 

The  preceding  would  present,  in  general,  a  problem  of 
extreme  difficulty,  especially  in  the  investigation  of  the  new 
limits  at  each  change  in  the  order  of  integration.  The  one 
matter  in  every  case  to  be  carefully  observed  is,  that  the  trans- 
formed integral  or  integrals  must  include  every  element 
which  enters  into  the  original   expression,   and  no  more. 

Again,  it  may  be  observed  that  in  the  foregoing  substitu- 
tions for  dx  dy  dz  the  order  may  be  interchanged  in  any 
manner. 

Thus,  if  we  commence  by  replacing  x  by  w,  we  must 
suppose  x  expressed  in  terms  of  u,y,z;  and  then  replace  dx 

by  -j-  du,  &c. 

J  du 


322  Multiple  Integrals. 

As  an  illustration  we  shall  consider  the  ordinary  trans- 
formation from  rectangular  to  polar  coordinates,  viz. : — 

x  =  r  sin  0  sin  0,     y  =  r  sin  9  cos  0,     z  =  r  cos  6. 

Here  we  have 

x*  +  y2  +  z2  =  r2 ; 

therefore  x2  =  r2  -  y2  -  z8 ; 


henoe 


<fe 


fl?r     a;     sin  0  sin  ^' 


Again  —  =  -  r  sin  0,      -f  =  -  r  sin  0  sin  0  ; 

au  aty 

..       .  d#  dz  dy       „    .     _ 

therefore  - — ^  -^  =  r2  sm  0 ; 

dr  dd  d$ 

and  for  the  element  of  volume  dx  dy  dz  we  substitute 

r2  Bin  6  dr  dO  d<p, 

a  result  which  can  be  also  readily  shown  from  geometrical 
considerations. 

Next,  let  us  consider  the  more  general  transformation 

x=r Binflv/i-m'siii8^,  y=r sin^^/i -n2sin20,  3=rcos0cos0, 
in  which  m2  +  nl  =  i. 

Squaring,  and  adding  the  three  equations,  we  get 

x2  +  y2  +  z2  =  r2. 
In  replacing  x  by  r,  we  get,  therefore, 
dx      r  l 


dr     x      sin  0  <f  i  -  m*  sin2  ^' 


Transformation  for  Implicit  Functions. 


32a 


Next,  to  replace  y  by  <j>,  we  must  express  y  in  terms  of  ry 
<j>,  and  2 :  thus 


y  =  r  sin  (j>  *Sm2  +  n2  cos3  0  =  sin  <f>   \, 


m'  r*  + 


COS2  (f> 


Hence 


tan  $  v/mVcos*  (p  +  n2  z2. 


m2  r2  sin2  (p 


—  =  sec2  d>  </m2  r2  cos8  <p  -t  n2  z2  -     ,  * 

d<p  T  r  i/mrr*  cos4  <p  +  n£  z* 

m2  r2  cos2  <j>  +  nzz2  sec2  <£      r  (m2  cos2  0  +  n2  cos2  0)  ( 
y'm2  r2  cos2  (p  +  n2z2        cos  0  y/m2  +  w2  cos2  0 ' 

and,  finally,  —  =  -  r  sin  0  cos  (p. 


Hence  for  dx  dy  dz  we  substitute 

r2  (m2  cos2  <p  +  n2  sin2  <p)  dr  d9  d<p 


v 


i  -  mr  sm 


in20  v^1  ~  ^2sin2 


(9) 


In  general  (ZW.  Calc.,  Art.  325),  the  product  - — i-  — 
v  dw  dv  du 

is  the  Jacobian  of  the  original  system  of  variables,  x,  y,  z, 

regarded  as  functions  of  the  new  system,  u,  v,  w. 

Accordingly,  the  general  substitution  for  dx  dy  dz  is 


dx 

dx 

dx 

du 

dv 

dw 

dy 
du 

dy 

dv 

dy 
dw 

dz 

dz 

dz 

du 

dv 

dw 

du  dv  dw. 


10) 


223.   Transformation  for  Implicit  Functions. — If, 

instead  of  being  given  x,  y,  z  explicitly  as  functions  of  w,  v,  w, 


324  Multiple  Integrals. 

we  are  given  equations  of  the  form 

Fx{x,y,zt  u,v,u>)  -o,  F,(«,y,i,  «*,v,w)  =o,  -Fl(a?,y,i,  w,»,w)  =  o, 

we  have  (D/^.  Cfl/c,  Art.  324),  adopting  the  usual  notation 
for  Jacobians, 

d(Fl9Ft,F9) 
d  (x,  y,z)  _        d  (u,  v,  w) 
d(ihv,w)        d{FltFt9F>y 
d  Qr,  //,  z) 
And  for  dx  dy  dz  we  must  then  substitute 

-=!  du  dv  dw,  (11) 

where  Jx  is  the  Jacobian  of  the  given  system  of  equations 
with  respect  to  the  new  variables,  and  Jz  their  Jacobian 
with  respect  to  the  original  system. 

224.  Transformation  of  Element  of  Surface. — If 

the  equation  of  a  surface  be  referred  to  a  system  of  rectangu- 
lar axes  it  is  easily  seen,  from  Art.  1 89,  that  the  element  of 
its  superficial  area,  whose  projection  on  the  plane  of  xy  is 
dx  dy,  is  equal  to 

Accordingly  the  area  of  a  surface  may  be  represented  by 


WJ1  +{j£W*Y* 


ixj     \dyj 

taken  between  proper  limits.  In  this  result  z  is  regarded  as 
a  function  of  x  and  y  by  means  of  the  equation  of  the 
surface. 

To  transform  this  expression  to  new  variables  u,  v,  we, 
by  the  preceding  Article,  substitute 

(  - — p  -  ~  —Adudv  instead  of  dx  dy. 
\du  dv     du  dv] 


General  Transformation  for  n  Variables. 


325 


Also 


du 


dz  dx      dz  dy 
dx  du      dy  du' 


dz      dz  dx      dz  dy 
dv      dx  dv      dy  dv 


therefore 


dz  dy 

dy  dz  "I 

dz 

m  du  dv 

du  dv 

dx 
dz 

dx  dy 
du  dv 

dz  dx 
dv  du 

dy  dx 
du  dv 

dx  dz 
dv  du 

dy 

dx  dy 
du  dv 

dy  dx 
du  dv 

(12) 


Substituting,  the  expression  for  the  superficial  area  be- 
comes 


\u 


dx  dy      dy  dx\r    (dx  dz      dz  dxV    (dz  dy      dy  dz  y 
dudv      dudv)     \dv  du      dvdu)     \dv  du      dv  du) 


22$.  General   Transformation   for   n  Variables. — 

The  transformation  of  Art.  223  can  be  readily  generalized. 
Thus,  for  the  case  of  n  variables,  in  the  transformation  of 
the  multiple  integral 

JJ7  .  .  Vdxi  dx%  dxz  .  .  .  dxn 

to  a  system  of  new  variables,  yl9  y29 .  .  .  ym  we  substitute 
for  dxi9  dxZ9  .  .  .  dxn  the  Jacobian  of  zl9  x2i  .  .  .  xn  regarded 
as  functions  of  yl9  y2,  y* . . .  y»  ;  hence 

dxx  dx2 .  .  .  dxn=j)  *'   2' '  "   \  dyx  dy2 .  .  .  dyn.       (13) 

And  in  the  case  of  implicit  functions,  we  substitute 
/a 


T  fyi  dy*  •  •  •  dVn, 


326 


Multiple  Integrals, 


where  Jx  and  J2  are  respectively  the  Jacobians  of  the  system 
of  equations  with  respect  to  the  new,  and  to  the  original, 
system  of  variables  (compare  Biff.  Calc,  Art  324). 

226.  Green's  Theorems. — We  shall  conclude  this 
Chapter  with  a  brief  notice  of  the  very  remarkable  theorems 
given  by  Green  ("  Essay  on  the  Application  of  Mathematics  to 
Electricity  and  Magnetism,"  Nottingham,  1828,  reprinted, 
1 871),  as  follows: — 

If  U  and  V  be  functions  of  x,  ?/,  z,  the  rectangular  coordi- 
nates of  a  point ;  then,  provided  U  and  V  are  finite  and  con- 
tinuous  for  all  points  within  a  given  closed  surface  S,  we  have 


1 


dUdV     dUdV     dUd 
dx    dx       dy   dy       dz    dz 


F) 


dx  dy  dz 


. 


where  the  triple  integrals  are  extended  to  all  points  within 
the  surface  8,  and  the  double  integrals  to  all  points  on  S ; 
aud  dn  is  the  element  of  the  normal  to  the  surface  at  dS, 
measured  outwards. 


For,  sinc< 
we  have 

3       £(#*Z\m*E 

dx\      dx  J     dx 

dV        d*V 
dx  +  U  dx'  ' 

- — -—dxdy  dz 
J  dx  dx 

*!J 

\U  -j-Ydxdydz,     (14) 

the  integrals  being  extended  to  all  points  within  8. 

Again,  since  S  is  a  closed  surface,  any  intersecting  right 
line  meets  it  in  an  even  number  of  points ;  consequently 


Green's  Theorems. 


327 


where  x^  #2,  27i,  TT2,  represent  the  values  of  x,  U,  for  two 
corresponding  points  of  intersection  with  8y  made  by  an 
indefinitely  thin  parallelepiped  standing  on  dydz;  and  2 
denotes  the  summation  extended  to  all  such  points  of  inter- 
section. Now,  as  in  Art.  192,  let  dSi9  dS2i  dS3,  &c,  repre- 
sent the  corresponding  elementary  portions  of  the  surface ; 
and  Xi,  X2,  X3,  &c,  the  angles  that  the  exterior  normals  make 
with  the  positive  direction  of  the  axis  of  x ;   we  shall  have 


dydz  =  cosXi  dSi 
Accordingly 
d 


cos  X2  dS2  =  cos  A3  dSt  =  -  cos  X4  dSi  =  &c. 


HI 


AU^ 


U  —  cos  X  dS. 
dx 


('5) 


under  the  same  restrictions  as  to  limits  as  before. 
Hence,  from  (14),  we  find 


jjfss?*** 


XI  —r-  cos  X  dS 
ax 


U  ~dtfdxdydz> 


along  with  corresponding  equations  for  y  and  z. 
Accordingly 


fdUdV  +dUdV     dUdV\ 
\dx  dx       dy  dy       dz   dz  J 


I 


r-fdV      .     dV  dV 

TJ   —  cosX  +  -7-  cosu  +  —  cos 
\dx  dy  dx 

TT(d*V    d*V    d*V\,    ,    . 


0 


dS 


Again,  we  obviously  have 

dx 
dn' 


cosX 


COS^u 


dy_ 
dn 


COS  V  = 


dn 


therefore         -7—  cos  X  +  -7—  cos  u  +  -7—  cos  v  =  -7— . 
a#  ay  dz  dn 


328 


Multiple  Integrals. 


Hence 

fdTJdV 


dUdV     dUdV\d 
dy  dy 


dz  dz 


d*V 

dx2 


(PV 
df 


d2V 
dz1 


dxdydz 


(16) 


1 

The  latter  expression  is  obtained  by  the  interchange  of  £7"  and 
V  in  the  preceding. 
If  ^=T,weget 


Ti*V    <?V  t  d2V 
dx2       dy2       dz2 


)  dxdydz. 


We  shall  now  determine  the  modification  to  be  made  when 
one  of  the  functions,  J7for  example,  becomes  infinite  within  S. 
Suppose  this  to  take  place  at  one  point  P  only :  moreover,  infi- 
nitely near  this  point  let  TJ  be  sensibly  equal  to  -,  r  being 

the  distance  from  P.  If  we  suppose  an  indefinitely  small 
sphere,  of  radius  a,  described  with  its  centre  at  P,  it  is  clear 
that  ( 1 6)  is  applicable  to  all  points  exterior  to  the  sphere ; 
also  since 

d*       d2       d2  \  i 


dx2  +  dy2  +  dz2)  r      °' 

it  is  evident  that  the  triple  integrals  may  be  supposed  to  extend 
through  the  entire  enclosed  space,  since  the  part  arising  from 
points  within  the  sphere  is  a  small  quantity  of  the  order  of  a2. 


Moreover,  the  part  of      U —  &8*  due  to  the  surface  of  the 
sphere  is  indefinitely  sn 
to  consider  the  part  of 


sphere  is  indefinitely  small  of  the  order  of  a.     It  only  remains 

V—  dS  due  to  the  spherical  sur- 


Examples.  329 

face.     Here,  as  V  is  supposed  to  vary  continuously,  we  may 
take  for  its  value  that  [V')  at  the  point  P :  also 

dU    dU    d\r)        i  _  _  i_ 

dn       dr  dr  r*         a2' 

consequently  the  value  of      V  —  dS,  for  the  sph 

-47rF'. 
Thus  (16)  becomes 

ffrr^^cv       fffrrA^       d*U       d^\7      7      7 


Lere  is 


17) 


where,  as  before,  the  integrals  extend  over  the  whole  volume 
and  over  the  whole  exterior  surface. 

The  same  method  will  evidently  apply  however  great 
may  be  the  number  of  points,  such  as  JP,  at  which  either  U 
or  V  becomes  infinite. 


Examples. 

r .  If  U=  a  cos  u  +  b  sin  u  cos  v  +  c  sin  wjsin  v , 

I     /( IT)  sin  u  dudv  =  2ir  I     f(Aw)  wdw, 
jo   Jo  J-i 


where  A=Va2  +  b*  +  c*. 

Let  x  =  cos  w,     y  =  sin  w  cos  v,     r  =  sin  w  sin  v  ; 

then   (#,  y,  z)  are  the  coordinates   of  a  point  on  a  sphere  of   unit  radius, 
with  centre  at  the  origin. 

Also  let  a  =  Aa,  b  =  A&,  c  =  Ay ;  then  a,  j9,  7  is  also  a  point  on  the  same 
sphere,  and 

a  cob  m  +  b  sin  «  cos  v  +  <;  sin  u  sin  v  =  -4  cos  0, 
[32] 


330  Multiple  Integrals. 

where  6  is  the  arc  joining  the  point  a,  $,  y  to  x,  y,  t.  Again,  the  element  of 
the  surface  of  the  sphere  at  the  latter  point  may  he  represented  by  sin  u  du  dv,  or 
by  sin  $  dQ  d<p,  indifferently.     Consequently 

/(acosttf  J  sin m cos  v  +  csinw  Bin  v)  sin  u  du  dv=f  (A  cob  6)  Bind  dd  d<l>. 

Integrating  each  of  these  over  the  entire  surface,  we  get 

rn(2w  C2ir[n  fir 

I*     /(U)amududv  =  \      \    f  {A  cob  $)  Bin  d  dd  d<p  =  2n  \  /(A  cos  6)  sin  6  d$. 

2.  Hence,  deduce  the  following  : 

I    I    f{U)  sinw  coBududv  =  — —      f(Aw)wdtv, 

fjrf2»r  2»<?f+1 

I        f(U)  sm  u  cob  vdudv=  — —      f(Aw)wdw. 
Jo  Jo  A  J_i 

These  are  deduced  from  (i)  by  differentiation  under  the  sign  of  integration. 

3.  Show  that  the  integral 

U  =  11  f  (*  +  y)  sM  ym'x  dx  dy, 

supposed  extended  to  all  positive  values  subject  to  the  condition  x  +  y  <  k,  ran 
be  reduced  to  a  single  definite  integral,  by  the  substitution 

x  =  uvy    y  =  w(i  —  v). 

Here  x  4-  y  =  u,  and  dx  dy  becomes  udv  dv  ;  also  the  limits  for  u  are  o  and  h ; 
and  those  for  v  are  o  and  1 ;  hence 

ftf1 


U=  I  f  f(u)  «'+»»- 1  v*-1  (1  -  v)"-ldudv 
J0J0 


r  (/)  r  (m)  f* 
"  rV+m)V(M) w,+m'1  *"'        ^""P"8 Art-  220- 

4.  Show  that  the  foregoing  process  can  be  extended  to  the  integral 
JJ  =  jjjf(x  +  y  +  2)  a;*"1  y»>-1  xn-i  dx  dy  dz,\ 
when  the  variables  are  always  positive  and  subject  to  the  condition 
.2 •  +  y  -f  z  <  a. 
Substitute  for  x  and  y  as  in  last ;  then,  regarding  z  as  constant,  the  limits 


Examples.  331 

for  v  are  o  and  I,  and  those  for  u  are  o  and  a  —  z ;  hence 

r(<  +  m)  Jo  Jo  ■"         ' 

=  rWrWr(«)r.  ,^ 

r  (£  +  m  +  »)  Jo  y  v  ; 


This  process  is  readily  extended  to  any  number  of  variables. 
5.  Find  the  value  of  the  definite  integral 


By  Art.  120  we  have 


Jo{*(i-v)+< 
Jo    Jo  «'#"» 


Transform  by  the  substitution  x=uv,  y  -  u  (1  -  v),  then,  sinoe  the  limits  for 
are  o  and  I,  and  those  for  u  are  o  and  00  ,  we  get 

«'&w  J0J0 

A   »W  (i-v)m-ldv 

therefore  f   '"(i-*)-1*    =      r(/)r(m) 

6.  Prove  that 

f      f      F{ax+bt/,a'x^b,y)dxdy  =  ^[      f      F(x,y)ixdy1 


where 

7.  Prove  that 


a'b' 


W      7T 

•  2  r  2    ( w2  cos2  0  +  «2  cos2  <p)ddd<p 


P2  r2     <m 
Jo  Jo  V(f 


w2sin20)(i-n2sin2</>)      2' 

when  m*  +  n2  =  1. 

This  is  an  immediate  consequence  of  (9),  Art.  222. 

8.  Show  that  Legendre's  Theorem  connecting  complete  elliptic  integrals 
with  complementary  moduli  follows  immediately  from  the  preceding  example 

[22a] 


332  Multiple  Integrals. 

It  IT 

J "5"  dd  f"5" 

o  Vl  -  m3  sin2  0  Jo 

then  the  result  given  in  Ex.  7  is  easily  transformed  into 

F(m)  E{n)  +  £(m)  F(n)  -  F(n)  F(m)  =  -. 
9.  Prove  that  the  area  of  a  surface  in  polar  coordinates  is  represented  by 


taken  between  suitable  limits. 

10.  Show  by  actual  integration  that 


cos  a  +  v  cos  0  +  w  cos  7)  dS, 

where  the  integrations,  respectively,  extend  through  the  volume  and  over  a 
closed  surface  S;  o,  £,  7  being  the  direction  angles  of  the  outward  drawn 
normal  at  dS. 

11.  Transform  the  multiple  integral 

J/J]"  Vdxdydzcko 
by  the  substitution 

x  =  r  cos  d  cos  <f>,    y  =  r  cos  0  sin  <f>,    z  =  r  sin  9  cos  if>,     to  =  r  sin  0  sin  ^. 

The  transformed  expression  is 

JUT  Fi  r3  sin  0  cos  0  rfrrf0  if  <% 

where  Fi  is  the  new  value  of  V. 

«a«8  «3«1  «1«2 

12.  11  %i  — ,     x%- ,    Xz  = , 

Ml  Mi  M3 

prove  that  jjj  Vdxy  dx%  dxz  transforms  into  4  J/J  Viduidu2du3. 


(  333  ) 


CHAPTER  XII. 

ON  MEAN  VALUE  AND  PROBABILITY. 

227.  A  very  remarkable  application  of  the  Integral  Calculus 
is  that  to  the  solution  of  questions  on  Mean  or  Average 
Values  and  Probability.  In  this  Chapter  we  will  consider  a 
few  of  the  less  difficult  questions  on  these  subjects,  which 
will  serve  to  give  at  least  some  idea  of  the  methods  em- 
ployed. We  will  suppose  the  student  to  be  already  acquaint- 
ed with  the  general  fundamental  principles  of  the  theory 
of  Probability. 

Mean  Values. 

228.  By  the  Mean  Value  of  n  quantities  is  meant  their 
arithmetical  mean,  i.e.  the  nth  part  of  their  sum. 

To  estimate  the  Mean  Yalue  of  a  continuously  varying 
magnitude,  we  take  a  series  of  n  of  its  values,  at  very  close 
intervals,  n  being  a  large  number,  and  find  the  mean  of  these 
values.  The  larger  n  is  taken,  and  consequently  the  smaller 
the  intervals,  the  nearer  is  this  to  the  required  mean  value. 

This  mean  value,  however,  depends  on  the  law  accord- 
ing to  which  we  suppose  the  n  representative  values  to  be 
selected,  and  will  be  different  for  different  suppositions. 
Thus,  for  instanoe,  if  a  body  fall  from  rest  till  it  attains  the 
velocity  v,  and  it  be  asked — What  is  its  mean  velocity 
during  the  fall  ?  If  we  take  the  mean  of  the  velocities  at 
successive  equal  infinitesimal  intervals  of  time,  the  answer 
will  be  \  v ;  but  if  we  consider  the  velocities  at  equal  intervals 
of  space,  it  will  be  f  v.  The  former  is  the  most  natural  sup- 
position in  this  case,  because  it  is  the  answer  to  the  question 
— What  is  the  velocity  with  which  the  body  would  move, 
uniformly,  over  the  same  space  in  the  same  time  ? — a  question 
which  implies  the  former  supposition.  We  might  frame  a 
similar  question,  of  a  less  simple  kind,  to  whioh  the  second 
value  above  would  be  the  answer. 


3o4  On  Mean  Value  and  Probability. 

Again,  if  we  wish  to  determine  the  mean  value  of  the 
ordinate  of  a  semicircle,  we  might  take  the  mean  of  a  series 
of  ordinates  equidistant  from  each  other ;  or  through  equi- 
distant points  of  the  circumference ;  or  suoh  that  the  areas 
between  each  pair  shall  be  equal :  in  each  case  the  mean 
value  will  be  different. 

Thus  we  see  that  the  Mean  Yalue  of  any  continuously- 
varying  magnitude  is  not  a  definite  term,  as  might  be  sup- 
posed at  first  sight,  but  depends  on  the  law  assumed  as  to  its 
successive  values. 

229.  Case  of  One  Independent  Variable. — We 
will  therefore  suppose  any  variable  magnitude  y  to  be  ex- 
pressed as  a  function  <p  (x)  of  some  quantity  x  on  which  it 
depends,  and  its  mean  value  taken  as  x  proceeds  by  equal 
infinitesimal  increments  h  from  the  value  a  to  the  value  b. 
Let  n  be  the  number  of  values,  then  nh  =  b  -  a.  The  mean 
value  is 


-  \<f>  (a)  +  0 (a  +  h)  +  <j>  {a  +  ih)  + 


But  (Art.  90), 

h  \<j>(a)  +  <j>(c 

1  +  h)  + 

0(a  +  2I1) 

Hence  the  mean  ' 

ralue  is 

M  = 

1 

b- 

a  \a  T 

<p(x)dx. 


(0 


Examples; 


1.  ,To  find  the  mean  value  of  the  ordinate  of  a  semicircle,  supposing  the 
s«ries  taken  equidistant. 

I     Cr        I 

M  =  —        Vr2  -  x*dx  -  - irr, 
2rJ.r  4 

viz.,  the  length  of  an  arc  of  450. 

2.  In  the  same  case,  let  us  suppose  the  ordinates  drawn  through  equidistant 
points  on  the  circumference. 

M  =  -  I    r  sin 0^0  =  -r ;  "the  ordinate  of  the  centre  of  gravity  of  the  arc. 
''Jo  T 


Case  of  One  Independent  Variable.  335 

3.  Determine  the  mean  horizontal  range  of  a  projectile  in  vacuo  for  different 
angles  of  elevation  from  450  -  0  to  450  +  0  ;  given  the  initial  velocity  V. 

If  a  be  the  angle  of  elevation,  the  range  is 

V* 

R  =  —  sin  2a. 
9 

1    f  F2 
Hence  M  =  —  I  —  sin  2ada,  between  the  limits  45 °  ±  0 ; 

20  J    g 

,       ,  m,     V2  sin  20 

therefore  M  = — . 

g  4   20 

2  V2 
The  mean  value  for  all  elevations,  from  o°  to  900,  is     — . 

7T   g 

4.  A  number  n  is  divided  at  random  into  two  parts  ;  to  find  the  mean  value 
of  their  product. 


M=-\    x(n  —  z)dz  =  -n2. 


5.  To  find  the  mean  distance  of  two  points  taken  at  random  on  the  circum- 
ference of  a  circle. 

Here  we  may  evidently  take  one  of  the  points,  A,  as  fixed, 'and  the  other,  B, 
to  range  over  the  whole  circumference :  since  by  altering  the  position  of  A  we 
should  only  have  the  same  series  of  values  repeated :  let  0  be  the  angle  between 
AB  and  the  diameter  through  A  :  as  we  need  only  consider  owe  of  the  two  semi- 
circles, 


2    f2 

M=-\ 


6.  To  find  the  mean  values  of  tbe  reciprocals  of  all  numbers  from  n  to  2», 
when  n  is  large  :  that  is,  to  find  the  mean  value  of  the  quantities 


n  n  n 

that  is,  the  mean  value  of  the  function  — ,  as  x  goes  by  equal  increments  from 

i  to  2 ; 

f  2  dx      I 
therefore  M=\    —  =  -log  2. 

Jx  nx     n 

7.  To  find  the  mean  values  of  the  two  roots  of  the  quadratic 

x2  —  ax  +  b  =  o, 

the  roots  being   known  to  be  real,  but  b  being. unknown,   except  that  it  is 
positive : 


336  On  Mean  Value  and  Probability. 

That  is,  b  is  equally  likely  to  have  any  value  from  o  to  —  ;  henoe  for  the 
greater  root,  o,  4 


i    r* 

r=r-2      adb 
i«2  Jo 


therefore  M  =  7  a. 

6 

The  mean  value  of  the  smaller  root  is  -  a. 

6 

The  mean  squares  of  the  two  roots  are  —  a2,   — a2.  These  might  be  deduced 
from  the  former  results,  since 

M (*2)  -  aM{x)  +  M{b)  =  o. 

8.  Find  the  mean  (positive)  abscissa  of  all  points  included  between  the  axis 
of  x  and  the  curve 

y  =  ae  r  .  .4ns.  — . 

Vir 
The  mean  square  of  the  abscissa  is  |c2. 

230.  If  Jf  be  the  mean  of  m  quantities,  and  M'  the  mean 
of  rri  others  of  the  same  kind,  and  if  /u  be  the  mean  of  the 
whole  m  +  m'  quantities,  we  have  evidently 

m3f+m'M'  .  . 

Thus  if  it  be  required  to  find  the  mean  distance  of  one  ex- 
tremity of  the  diameter  of  a  semicircle  from  a  point  taken  at 
random  anywhere  on  the  whole  periphery  of  the  semicircle ; 
since  the  mean  value  when  it  falls  on  the  diameter  is  r,  and 

the  mean  value  when  it  falls  on  the  arc  is  — ,  we  have 


4r 
2r  .  r  +  nr —  , 
7T  6r 

2v  +  7rr  2  +  tt 


Case  of  Two  or  More  Independent  Variables.  337 

231.  Case  of  Two  or  More  Independent  Variables. 

— If  z  =  <p  {x,  y)  be  any  function  of  two  independent  variables, 
and  x,  y  be  taken  to  vary  by  constant  infinitesimal  increments 
h,  k,  between  given  limits  of  any  kind,  the  mean  value  of  the 
function  z  will  be 

Jjzdxdy 

\\dxdy>  {i} 

both  integrals  being  taken  between  the  given  limits. 

The  easiest  way  of  seeing  this  is  to  suppose  xy  y,  z  the 
coordinates  of  a  point ;  and  to  conceive  the  boundary,  repre- 
senting the  limits,  traced  on  the  plane  of  xy,  and  then  ruled 
by  lines  parallel  to  x,  y  at  intervals  k,  h  apart.  We  have 
thus  a  reticulation  of  infinitesimal  rectangles  hk ;  and  if  at 
each  angle  an  ordinate  z  be  drawn  to  the  surface  z  =  (f>(x,  y), 
as  the  number  of  ordinates  will  be  the  same  as  that  of  rect- 
angles, we  shall  have 

volume  jjzdxdy  =  sum  of  ordinates  x  hk ; 

also  the  plane  area  jj  dxdy  =  number  of  ordinates  x  hk ; 

so  that  dividing  the  sum  of  the  ordinates  by  their  number, 
the  above  expression  results. 

It  may  be  shown,  in  like  manner,  that  for  three  or  more 
independent  variables  a  similar  expression  holds. 

It  is  evident  that  the  above  expression,  viewed  geometri- 
cally, gives  the  mean  value  of  any  function  of  the  coordinates 
of  a  series  of  points  uniformly  distributed  over  a  given  plane 
area. 

Examples. 

1.  Suppose  a  straight  line  a  divided  at  random  at  two  points,  to  find  the 
average  value  of  the  product  of  the  three  segments. 

Let  the  distance  of  the  two  points  X,  Y,  from  one  end  A  of  the  line,  he 
called  x,  y.  Consider  first  the  cases  when  x  >  y  ;  the  sum  of  the  products  for 
these  is  half  the  whole  sum  ;  hence 

M=~%\  j  y (* - y) (« - x) dxdy  =  ■^>a3- 

2.  A  numher  a  is  divided  into  three  parts ;  to  find  the  mean  value  of  one 
part. 


338 


On  Mean  Value  and  Probability. 


Let  z,  y,  a  —  x  —  y,  be  the  parts  ; 

ra  ra-x 

\  xdxdy 

Jo  Jo  I 


M 


J  a  ra-z 


-a. 
3 


This  value  might  be  deduced,  without  performing  the  integrations,  by  consider- 
ing that  the  expression  is  the  abscissa  of  the  centre  of  gravity  of  the  triangle 
OA.B  ;  OA,  OB  being  lengths  taken  on  two  rectangular  axes,  each  =  a. 

Of  course  the  result  in  this  case  requires  no  calculation ;  as  the  sum  of  the 
mean  values  of  the  three  parts  must  be  =  a ;  and  the  three  means  must  be  equal. 

The  mean  square  of  a  part  is  -  a2. 

3.  A  number  a  is  divided  at  random  into  three  parts :  to  find  the  mean 

value  of  the  least  of  the  three  parts:    also  of  the  greatest,  and  of  the  mean. 

Let  x,  y,  a  -  x  —  y,  be  the  greatest,  mean,  and  least  parts.     The  mean  value 

of  the  greatest  iaM=  {.,      '    :   the  limits  of  both     - 
tidxdy 

integrations  being  given  by 

x>y>a-x-y>o. 

If  x,  y  be  the  coordinates  of  a  point,  referred 
to  the  axes  OA,  OB,  taking  OA  =  OB  =  a,  the 
above  limits  restrict  the  point  to  the  triangle  A  VH 
{AM  being  drawn  to  bisect  OB)  ;  and  the  above 
value  of  M  is  the  abscissa  of  the  centre  of  gravity  of 

this  triangle ;  i.  e.  -  of  the  sum  of  the  abscissas  of  its 

angles;  hence 

„     !  /        1         I    \       II 

M  =  -[a+     a  +-a]  =  —  a. 
3  \        2         3/1* 

The  ordinate  of  the  same  centre  of  gravity,  viz., 


Fig.  S3- 


3\2         3    /       li 


is  the  mean  value  of  the  mean  part ;  hence  the  mean  values  of  the  three  parts 
required  are  respectively 


4.  To  find  the  mean  square  of  the  distance  of  a  point  within  a  given  square 
(side  =  2a),  from  the  centre  of  the  square. 


4«    J-a  J-a  3 


Case  of  Two  or  More  Independent  Variables.  339 

It  is  obvious  that  the  mean  square  of  the  distance  of  all  points  on  any  plane 
area  from  any  fixed  point  in  the  plane  is  the  square  of  the  radius  of  gyration  of 
the  area  round  that  point. 

5.  To  find  the  mean  distance  of  a  point  on  the  circumference  of  a  circle  from 
all  points  inside  the  circle. 

Taking  the  origin  on  the  circumference,  and  the  diameter  for  the  axis,  if  dS 
be  any  element  of  the  area,  we  have 


M=J- — —  =  —  r2dddr  =  - — . 


97T 


232.  Many  problems  on  Mean  Values,  as  well  as  on 
Probability,  may  be  solved  by  particular  artifices,  which,  if 
attempted  by  direct  calculation,  lead  to  difficult  multiple 
integrals  which  could  hardly  be  dealt  with. 

Examples. 

1.  To  find  the  mean  distance  between  two  points  within  a  given  circle. 

If  M  be  the  required  mean,  the  sum  of  the  whole  number  of  cases  is  repre- 
sented by 

(*r2)2J/. 

Now  let  us  consider  what  is  the  differential  of  this,  that  is,  the  sum  of  the  new 
cases  introduced  by  giving  r  the  increment  dr.  If  M0  be  the  mean  distance  of 
a  point  on  the  circumference  from  a  point  within  the  circle,  the  new  cases  intro- 
duced by  taking  one  of  the  two  points  A  on  the  infinitesimal  annulus  nrrdr,  are 

irr2  M0  .  2-irrdr  ; 

doubling  this,  for  the  cases  where  the  point  B  is  taken  in  the  annulus,  we  get 
d.  {{irr^M}  =4TT*M0r*dr. 

Now  M0  =  —  (Ex.  5,  Art.  231)  ; 
9* 

therefore  •tflr^M— it       r*dr; 

9        .'0 

12S 
therefore  M  = r. 

2.  To  find  the  mean  square  of  the  distance  between  two  points  taken' on  any 
plane  area  £1. 

Let  dS,  dS'  be  any  two  elements  of  the  area,  A  their  mutual  distance,  and 
we  have 

M=~^A^dSdS\ 


340 


On  Mean  Value  and  Probability. 


Now,  fixing  the  element  dS,  the  integral  of  A*dS'  is  the  moment  of  inertia 
of  the  area  n  round  dS ;  so  that  if  K  =  radius  of  gyration  of  the  area  round  dS, 

M=-JSK>dS: 

let  r  =  distance  of  dS  from  the  centre  of  gravity  G  of  the  area,  k  the  radius  of 
gyration  round  G ;  then 

Z2  =  r2  +  F  : 


therefore 


M=k*  +  -jjr*dS=2k*; 


thus  the  mean  square  is  twice  the  square  of  the  radius  of  gyration  of  the  area 
round  its  centre  of  gravity. 

233.  The  mean  distance  of  a  point  P  within  a  given  area 
from  a  fixed  straight  line  (which  does  not  meet  the  area)  is 
evidently  the  distance  of  the  centre  of  gravity  G  of  the  area 
from  the  line.  Thus,  if  A,  B  are  two  fixed  points  on  a  line 
outside  the  area,  the  mean  value  of  the  area  of  the  triangle 
APB  =  the  triangle  AGB. 

From  this  it  will  follow,  that  if  X,  Y9  Z  are  three  points 
taken  at  random  in  three  given  spaces  on  a  plane  (such  that 
they  cannot  all  be  cut  by  any  one  straight  line),  the  mean 
value  of  the  area  of  the  triangle  XYZ  is  the  triangle  GG'G", 
determined  by  the  three  oentres  of  gravity  of  the  spaces. 

Example. 

1.  A  point  P  is  taken  at  random  within 
a  triangle  ABC,  and  joined  with  the  three 
angles.  To  find  the  mean  value  of  the 
greatest  of  the  three  triangles  into  which 
the  whole  is  divided. 

Let  G  be  the  centre  of  gravity ;  then  if 
the  greatest  triangle  stands  on  AB,  P  is 
restricted  to  the  figure  CHGK,  and  the 
mean  value  of  APB  is  the  same  as  if  P 
were  restricted  to  the  triangle  GCK;  hence 
we  have  to  find  the  area  of  the  triangle 
whose  vertex  is  the  centre  of  gravity  of 
GCK,  and  base  AB  ; 


Fig.  54- 


therefore  M  =  -  {ACB  +  AKB  +  AGB)  =  -  ( 1  +  -  +  -\  ABC ; 

hence  the  mean  value  is  —  of  the  whole  triangle. 

Io 

The  mean  values  of  the  least  and  mean  triangles  are  respectively  -  and 

9         io 

of  the  whole. 

This  question  can  readily  be  shown  to  be  reducible  to  Question  3,  Art.  231. 


Case  of  Two  or  More  Independent  Variables.  341 

234.  If  M  be  the  mean  value  of  any  quantity  depending 
on  the  positions  of  two  points  (e.  g.  their  distance)  which  are 
taken,  one  in  a  space  J/,  the  other  in  a  space  B  (external  to 
A)  ;  and  if  M '  be  the  same  mean  when  both  points  are  taken 
indiscriminately  in  the  whole  space  A  +  B ;  MA ,  MB  the 
same  mean  when  both  points  are  taken  in  A,  or  both  in  B, 
respectively;  then 

(A  +  B)2 Mf -  2 ABM  +  A2MA  +  B*MB.  (4) 

If  the  space  A  =  B, 

4M'=2M+  MA  +  MB; 

if,  also,  MA  =  MM, 

2Mr=M+MA; 

thus  if  M  be  the  mean  distance  of  a  point  within  a  semi- 
circle from  one  in  the  opposite  semicircle,  Mi  that  of  two 
points  in  one  semicircle,  we  have  (Art.  232) 

M+M1  =  ^r. 
45T 

To  determine  M  or  Mx  is  rather  difficult,  though  their 

sum  is  thus  found.     The  value  of  M  is -=  r. 

Examples. 

1.  Two  points  X,  Fare  taken  at  random  within  a  triangle.  What  is  the 
mean  area  M  of  the  triangle  XYC,  formed  by  joining  them  with  one  of  the 
angles  of  the  triangle  ? 

Bisect  the  triangle  by  the  line  CD  ;  let  M\  be  the  mean  value  when  both 
points  fall  in  the  triangle  ACD  ;  Mz  the  value  when  one  falls  in  ACB  and  the 
other  in  BCD ;  then 

2M=Mi  +  Mi. 

But  Jfi  =  -M ;  and  Mz  =  GG'C,  where  G,  G'  are  the  centres  of  gravity 

2 

of  ACD,  BCD,  this  being  a  case  of  the  theorem  in  Art.  233  ;  hence 

M2  =  -ABC,    and     M=^-ABC. 
9  27 

2.  To  find  the  mean  area  of  the  triangle  formed  by  joining  an  angle  of  a 
square  with  two  points  anywhere  within  it. 


X       /, 

Y 


342  On  Mean  Value  and  Probability. 

By  a  similar  method  this  is  found  to  he 

-=-r  of  the  whole  square. 

3.  What  is  the  mean  area  of  the  triangle  formed  hy  joining  the  same  two 
points  with  the  centre  of  the  square  ? 

We  may  take  one  of  the  points  X  always  in  the  square  OA ;  take  the  whole 
square  as  unity  ;  then  if  M  be  the  mean,  the  sum     u  n 

01  all  the  cases  is 

-  M  =  -=  Mi  +  2  -  M 2  +  -  M s, 
4  42  42  42 

Mu  M2,  Mz  being  the  mean  areas  when  the  second 
point  Y  is  taken  respectively  in  OA,  OB,  and  OC. 
But  Ms  =  M\,  for  to  any  point  Y  in  00  there  cor- 
responds one  Y'  in  OA,  which  gives  the  area 
OXY'  =  OXY; 

therefore  M  =  -  Mi  +  -  M2. 

2  2  Fig.  55. 

But  Mi  =  —0.-,    M2  =  —;  hence  M=  --  of  the  whole  square.* 
108   4  16  108  * 

235.  If  two  spaces  A  +  C,  B  +  C  have  a  common  part  C, 
and  M  be  any  mean  value  relating  to  two  points,  one  in  A  +  C, 
the  other  in  B  +  C ;  and  if  the  whole  space  A  +  B  +  C  =  W, 
and  31 w  be  the  same  mean  when  both  points  are  taken  indis- 
criminately in  W;  MA  when  taken  in  A,  &c,  then 

2(A  +  C)(B+C)M=W2Mw+C2Mc-A2MA-B*MJi,     (5) 

as  is  easily  seen  by  dividing  the  whole  number  W2  of  cases 
into  the  different  classes  of  cases  which  compose  it. 


*  In  such  questions  as  the  above,  relating  to  areas  determined  by  points 
taken  at  random  in  a  triangle  or  parallelogram,  we  may  consider  the  triangle  as 
equilateral,  and  the  parallelogram  as  a  square.  This  will  appear  from  orthogonal 
projection ;  or  by  deforming  the  triangle  into  a  second  triangle  on  the  same 
base  and  between  the  same  parallels,  when  it  is  easy  to  see  that  to  one  or  more 
random  points  in  the  former  there  correspond  a  bike  set  in  the  latter,  determining 
the  same  areas.  This  second  triangle  may  be  made  to  have  a  side  equal  to  a 
Bide  of  an  equilateral  triangle  of  the  same  area ;  and  then  be  deformed  in  like 
manner  into  the  equilateral  triangle  itself.  Likewise  a  parallelogram  may  be 
deformed  into  a  square. 


Case  of  Two  or  More  Independent  Variables.  343 


Example. 

Two  segments,  AB,  CD,  of  a  straight  line  have  a  common  part  CB;  to 
find  the  mean  distance  of  two  points  taken,  one  in  AB,  the  other  in  CD. 

lAB  .  CD.  M=AD* .-  AD  +  CBK-CB-ACP.-AC-BD^.-BD, 
3  3  3  3 

since  the  mean  distance  of  two  points  in  any  line  is  -  of  the  line  ; 

AD3  +  CBS  -  AC*  -  DBZ 
therefore  M= ZaWTCD 

236.  The  consideration  of  probability  often  may  be  made 
to  assist  in  determining  mean  values.  Thus,  if  a  given 
space  8  is  included  within  a  given  space  A,  the  chance  of  a 
point  P,  taken  at  random  on  A,  falling  on  S,  is 

8 

P  =  A' 


But  if  the  space  S  be  variable,  and  M[8)  be  its  mean  value, 

,-Z®.  (6) 

For,  if  we  suppose  S  to  have  n  equally  probable  values 
Si,  S2,  Sz  .  .  .  .,  the  chance  of  any  one  5i  being  taken,  and  of 
P  falling  on  &,  is 

P^nAl 

now  the  whole  probability  p  =  px  +  p2  +  p3  +  .  .  . ;  which  leads 
at  once  to  the  above  expression. 

The  chance  of  two  points  falling  on  S  is 

P  =  -^'  (7) 

In  such  a  case,  if  the  probability  be  known,  the  mean  value 
follows,  and  vice  versa.  Thus,  we  might  find  the  mean  value 
of  the  distance  of  two  points  X,  Y  taken  at  random  in  a  line, 


344  On  Mean  Value  and  Probability. 

by  the  consideration  that  if  a  third  point  Z  be  taken  at  random 

in  the  line,  the  chance  of  it  falling  between  X  and  Y  is  -  ;  as 

one  of  the  three  must  be  the  middle  one.     Hence  the  mean 

distance  is  -  of  the  whole  line. 
3 

2an 


Again,  the  mean  nth  power  of  the  distanoe  is 


[n+i)(n  +  2)f 

where  a  -  whole  line.     For  if  p  is  the  probability  that  n  more 
points  taken  at  random  shall  fall  between  X  and  F, 

M{XY)n=anp. 

Now  the  chance  that  out  of  the  n  +  2  points,  X  shall 

2 
be  one  of  the  extreme  points  is ;   and  if  it  is  so,  the 

n+  2 

chance  that  Y  shall  be  the  other  extreme  point  is . 

n  +  1 

Examples. 

1.  From  a  point  X  taken  anywhere 
in  a  triangle,  parallels  are  drawn  to  two 
of  the  sides.  Find  the  mean  value  of 
the  triangle  TJXV. 

If  a  second  point  X'  be  taken  at 
random  within  ABC,  the  chance  of 
its  falling  in  XUV  is  the  same  as  the 
chance  of  X  falling  in  the  correspond- 
ing triangle  X'  U'  V ;  that  is,  of  X' 
falling  on  the  parallelogram  XC.  Hence 
the  mean  value  of  UX  V  =  mean  value 
of  XC.     But  the  mean  value  of  ( UX  V 

+  XC)  is  -  ABC;  as  the  whole  triangle 

o 

can  he  divided  into  three  such  parts  hy  drawing  through  X  a  parallel  to  AB.  * 
Thus 

M(UXV)  =  ^ABC. 
6 

The  mean  value  of  UV  ia-AB.    For  TJV  is  the  same  fraction  of  AB  that  the 

3 
altitude  of  X  is  of  that  of  C :  see  Art.  233. 

*  The  triangle  may  be  considered  equilateral :  see  note,  Art.  234. 


Case  of  Two  or  More  Independent  Variables.  345 


Cor.  Hence,  if  p  1 
triangle  ABC,  we  get 


the  perpendicular  from  X  on  AB,  h  the  altitude  of 


M(?)  =  -h* 


If  the  area  ABC  he  taken  as  unity,  we  have,  since  J7XV:  AXB=AXB  :  ABC, 
{AXB)*=UXV. 

Thus  the  mean  square  of  the  triangle  AXB  is  -.    If  two  other  points  Y,  Z  are 

taken  at  random  in  the  triangle,  the  chance  of  hoth  falling  on  AXB  is  thus  the 

same  as  that  of  a  single  point  falling  on  TJXV  \  i.  e.  -.    Hence  we  may  easily 

6 

infer  the  following  theorem  :— 

If  three  points  X,  Y,  Z  are  taken  at  random  in  a  triangle,  it  is  an  even 
chance  that  Y,   Z  hoth  fall  on  one  of  the  triangles 
AXB,  AXC,  BXC.  D 

2.  In  a  parallelogram  A  BCD  a  point  X  is  taken  at 
random  in  the  triangle  ABC,  and  another  Y  in  ABC. 
Find  the  chance  that  X  is  higher  than  Y. 

Draw  XH  horizontal :  the  chance  is 

mean  area  of  AMK  -f-  ADC. 

But  AHK=XUV,  and  the  mean  area  of  XUV=  \  ACB   H 
i  6 

(Ex.  i) ;  hence  the  chance  is  ^. 

■A 

3.  If  0  he  a  point  taken  at  random  on  a  triangle,  and 

lines  he  drawn  through  it  from  the  angles,  to  find  the 
mean  value  of  the  triangle  JDEF.     (Mr.  Miller.) 

It  will  he  sufficient  to  find  the  mean  area  of  the  triangle  AEF,  and  subtract 
three  times  its  value  from  ABC.  If  we  put  a,  |8,  7  for  the  triangles  BOO, 
AOC,  AOB,  it  is  easy  to  prove 


AEF= 


07 


(a  +  0)  (a  +  7) 


ABC. 


If  we  put  the  whole  area  ABC 
he  the  element  of  the  area  at  0, 


1,  and  if 


'^-\\(^?= 


7)' 


Fig.  58. 


the  integration  extending  over  the  whole  triangle. 

But  if  p,  q  are  the  perpendiculars  from  0  on  the  sides  b,  e,  it  may  he  easily 
shown  that  the  element  of  the  area  is 


dpdq 
sin  A 


be  sin  A         '  ' 

[23] 


346  On  Mean  Value  and  Probability. 

Thus  the  mean  value  of  A  EF  becomes 

Again,  by  Art.  95,  the  definite  integral 
f^logjS 


0    I  -  «  6 


therefore  Jf  =  -  1  -  2  [  1  —  —  )  =  —  -  *. 

\       fit       3      * 


7T2 

3 
Hence  the  mean  value  of  the  triangle  BEF  is 

10  -  7T2, 

that  of  ABC  being  unity. 

It  is  curious  that  the  same  value,  10  -  tt2,  has  been  found  by  Col.  Clarke  to 
be  the  mean  area  of  a  triangle  formed  by  the  intersections  of  three  lines,  drawn 
from  A,  B,  C  to  points  taken  at  random  in  a,  b,  c  respectively. 

4.  To  find  the  average  area  of  all  triangles  having  a  given  perimeter  (2*). 
By  this  is  meant  that  the  given  perimeter  is  divided  at  random  in  every  possible 
way  into  three  parts,  a,  b,  c,  and  only  those  cases  are  taken  in  which  a,  b,  c  can 
form  a  triangle  ;  then  the  mean  value  of 

A  =  ^/  #(•-•)  (t~»)(#^4  A  i  \  b 

Fig.  59- 
has  to  be  found. 

|f  H  Take  AB  =  2#,  let  X,  Y  be  the  two  points  of  division,  AX  =  x,  AY=y: 
these  are  subject  to  the  conditions 

x  <  s,     y>s,     y-x<». 

A  

*ow  ~7J  =  </('  -  *)  (y  - «)  («  "  V  +  *)  J 

1  [T  v7 (•  -  *)  (y  -«)(«-  y  +  *) •  <*y  rf* 

Again,  by  Art.  132,  we  have 


I" 


y/(s  -  x)  (s  -  y  +  *)  dx  =  '    (2*  -  yf ; 


The  result  is  therefore  :— Mean  area  =  —  (2*)2. 
In  the  same  case  we  should  easily  find 

Mean  square  of  area  =  — . 


Case  of  Two  or  More  Independent  Variables.  347 

5.  Three  points  axe  taken  at  random  within  a  given  triangle ;  prove  that  the 
mean  area  of  the  triangle  formed  by  them  is  —  of  the  given  triangle. 

Call  the  area  of  the  given  triangle  A,  the  required  mean  M :  we  will  first 
prove  that  if  M0  be  the  mean  area  when  one  of  the  three  points  is  restricted  to  a 
side  of  the  given  triangle, 

M=-M0. 

4 

Let  A  receive  an  increment  of  area  dA,  by  adding  to  it  an  infinitesimal  band 
included  between  the  base  a  and  a  line  parallel  to  it ;  the  increase  produced  in 
the  sum  of  all  the  cases  is  found  by  considering  one  of  the  random  points  X 
taken  in  this  band  ;  the  additional  cases  introduced  will  be 

A'2dA .  M0. 

The  whole  increase  is  treble  this,  for  we  must  consider  also  the  cases  when 
Y,  Z  fall  in  this  band  (the  cases  when  tivo  of  the  three  fall  on  it  may  be 
neglected,  their  number  being  proportional  to  the  square  of  dA).  Now  the  sum 
of  all  the  original  cases  is  A3M ;  hence 

d(A3M)  =SA2ModA. 

M 
Now  —  is  constant  for  all  triangles  (see  note, 

A 
Art.  234) ; 

hence  —  d .  A*  =  3A2M0dA ;     .-.  M  =  -  M0. 
A  4 


Again,  to  find  Jfo,  consider  the  random  point  X  fixed  at  a  particular  point 
iTof  the  base  a,  the  other  two  points,  T,  Z,  ranging  all  over  the  triangle.  Let 
M'  be  the  mean  value  of  BYZ;  the  sum  of  all  the  cases,  viz.,  A2M',  may  be 
decomposed  into  three  groups :  (1)  when  T,  Zaxe  in  ABB\  (2)  both  in  ACD  ; 
(3)  one  in  each  triangle  : 

.-.  {ABCfW  =  {ABLf.  4  ABD+  (ACJD)*  .  -^ACD  +  zABD .ACX>.^?f 

by  Ex.  (1),  Art.  234,  and  because  in  case  (3)  the  mean  value  is  the  area  of  the 
triangle  formed  by  joining  D  with  the  centres  of  gravity  of  ABB  and  ACB 
(Art.  233).     Let  BB  =  z,  altitude  of  triangle  =  p,  and  we  get 

Now  when  the  point  X  falls  in  the  element  dx,  the  sum  of  all  the  cases  is 
[23a] 


348  On  Mean  Value  and  Probability. 

ArM'dx ;  and  hence,  when  X  ranges  from  B  to  C,  the  whole  sum  of  cases  is 

Jo  jq\21        27  9  ) 

therefore  «AW0  =  (i»)3  -  a4  =  -  a  A3. 

9        9 

Hence  Mq  =  -  A';  and  therefore  M  -  -  M0  =  —  A. 
9    '  4  12 

Cob.  Hence,  if  four  points,  A,  B,  C,  D,  are  taken  at  random  within  a 

triangle,  the  chance  that  they  determine  a  re-entrant  quadrilateral  is  -•    For  • 

the  chance  that  D  falls  in  ABC  is  the  mean  value  of  ABC  divided  by  the 

whole  triangle,  that  is  —  ;  and  we  have  to  add  to  this  the  chances  that  C  falls 

2 

in  ABB,  &c.     The  chance  that  ABCD  is  convex  is  -. 

6.  The  mean  distance  of  the  vertex  of  a  triangle  from  all  points  in  the  area  is 

Xal  to  its  distance  from  the  centre  of  gravity,  measured  along  a  parabolic  path, 
ch  leaves  the  vertex  in  the  direction  of  one  of  its  sides,  and  reaches  the 
centre  of  gravity  in  a  direction  parallel  to  the  other— the  axis  of  the  parabola 
being  parallel  to  the  base. 

Let  an  indefinite  line  AP  be  con-  ^ 

ceived  to  revolve  round  A,   from  the  \fr    yj^. 

direction  AC  to  AB  ;  and  as  it  revolves,  •  "*     /I \\\ 

suppose  that  all  the  mass  of  the  triangle  $6/*{  \  \\\ 

ABC  which  lies  to  the  right  of  it  is  tJ^  v'\    X\  ^\ 

transferred  continuously  to  the  vertex^.  *■/""" f\         \\      \^ 

The  centre  of  gravity  of  the  whole  mass  /  \^     \\         \ 

will  thus  describe  a  curve  starting  from         /  q     \\        \. 

G,  and  ending  at  A.    When  the  line  is       /  \\  \^ 

at  AP  let  the  centre  of  gravity  be  at  g  ;  -jf '^  V\ _\ 

and  when  it  is  in  the  consecutive  position  m        V  J*    P  0 

AP',  let  fhe  centre  be  at  /.     As  the  Fig-  6i. 

mass  of  the  triangle  A PP'  has  been  transferred  to  A,  gg'  is  parallel  to^P;  also 

,     APP'     2  A_ 
yy       ABC      3 

2 

since  -  AP  is  the  distance  traversed  by  the  centre  of  gravity  of  the  transferred 
portion  of  the  whole  mass.* 

2 

But  as  -  AP  is  the  mean  distance  of  all  points  in  APP'  from  A,  the  sum  of 

2 

every  element  in  APP'  into  its  distance  from  A  =  APP'  x  -  AP.    Hence  the 

sum  of  all  the  elements  gg',  i.  e.  the  whole  arc  GA  =  sum  of  every  element  of 
ABC  into  its  distance  from  A,  divided  by  the  area  ABC,'\.  e.  the  mean  distance 
required. 

*  See  Rankine,  Applied  Mechanics,  p.  54. 


Probabilities,  349 

It  is  eaay  to  show  that  if  gT  is  drawn  parallel  to  BO, 

AT*  =  ^-gT; 
3« 

so  that  the  curve  is  the  parahola  mentioned  ahove.  For  A  and  g  are  in  directum 
with  the  centre  of  gravity  of  ABP;  and  hence,  as  g  is  the  centre  of  gravity  of 
ABP  and  a  mass  at  A  equal  to  APC, 

AT  _BP  BP_  _e_ 

—  -— ,  and      T-AT- 

—  c 

3 


PROBABILITIES. 

237.  The  calculation  of  Probabilities,  when  the  number 
of  favourable  cases,  as  well  as  the  whole  number  of  cases,  is 
finite,  is  not  a  subjeot  for  the  Infinitesimal  Oaloulus.  It  is 
when  the  number  of  cases  depends  on  continuously  varying 
magnitudes,  and  is  therefore  infinite,  that  recourse  has  to  be 
made  to  the  methods  of  the  Integral  Calculus. 

The  same  remark  applies  here  which  we  had  occasion  to 
make  as  to  mean  values  (Art.  228).  The  value  of  the  pro- 
bability will  depend  on  the  law  according  to  whioh  we  select 
the  series  of  cases  which  we  take  as  representing  the  total 
number — that  is,  it  will  depend  on  which  variable  (or  varia- 
bles) we  suppose  to  be  taken  at  random,  that  is,  to  proceed  by 
constant  infinitesimal  increments  ;*  in  other  words,  to  be  the 
independent  variable  (or  variables).  Thus,  if  we  have  to  find 
the  chance  of  the  line,  drawn  from  a  fixed  point  to  a  given 
finite  straight  line,  exceeding  a  given  length,  the  results  will 
be  different  if,  first,  we  suppose  a  series  of  lines  drawn  to 
points  taken  at  random  on  the  given  line,  or,  seoondly,  a 
series  of  lines  drawn  in  random  directions  from  the  fixed 
point.  In  many  cases,  however,  the  problem  has  an  obvious 
sense  which  precludes  any  such  uncertainty. 

238.  Let  us  consider  a  simple  question  on  chances.  Two 
integers  are  ohosen  at  random  from  o  to  6  inclusive  ;  to  find 


*  Of  course  a  large  numher  of  values  taken  at  random  for  a  variable  do  not 
really  form  an  equi- different  series  :  but,  as  they  must  give  a  number  of  points 
(when  measured  along  a  straight  line)  of  uniform  density,  they  may  be  taken, 
for  the  purposes  of  calculation,  as  equi-different. 


350  On  Mean  Value  and  Probabilit  >/ . 

the  chance  that  the  greater  of  the  two  exceeds  a  given  value, 
suppose  3.  Here  the  whole  number  of  cases,  all  equally 
probable,  is  easily  seen  to  be 

1  +  2  +  3  +  4  +  5  +  6, 

and  the  number  of  favourable  cases  is 

4  +  5  +  6, 

so  that  the  required  chance  is  -. 

If,  however,  the  question  is  not  confined  to  integers,  but 
the  two  numbers  chosen  may  have  any  arbitrary  values  from 
o  to  6  ;  or  as  we  may  state  the  question  : — Two  quantities 
are  taken  at  random  from  o  to  a ;  find  the  chance  that  tho 
greater  of  the  two  is  less  than  a  given  value  b  : — 

Let  x  be  the  greater;  then  for  any  assigned  value  of  x 
the  number  of  cases  is  measured  by  x  (since  the  lesser  may  have 
any  value  from  o  to  x) ;  henoe  the  number  of  cases  when  the 
greater  falls  between  x  and  x  +  dx  is  measured  by  xdx ;  tho 


whole  number  of  cases  is  therefore 


xdx ;  and  the  favourable 


Cb  .  b2 

cases  are  I    xdx.    The  required  chance  is  therefore  p  =  —2. 

This  instance  will  serve  to  show  how  the  Integral  Calculus 
may  enter  into  the  estimation  of  chances.  It  is  true  that  it 
might  easily  be  solved  otherwise  ;  for  if  the  two  numbers  are 
considered  as  the  distances  of  two  points  taken  at  random  in 
a  line  of  length  0,  from  one  end  of  the  line,  and  if  we 
measure  a  distance  b  from  that  end,  the  problem  is  really  to 
find  the  chance  that  both  points  fall  within  b  ;  which  chance 

is  evidently  — ' 

239.  We  prooeed  to  give  a  few  easy  questions  on  proba- 
bilities :  general  rules  can  hardly  be  given  for  their  solution, 
the  number  and  diversity  of  the  questions  which  may  be 
proposed  being  so  great  that  no  attempt  seems  to  have  been 
made  to  classify  or  conneot  them  into  a  regular  theory.  We 
will  give,  in  particular,  several  on  Looal  or  Geometrical 
Probability. 


Probabilities. 


351 


Examples. 

i.  If  an  event  B  is  known  to  have  occurred  in  a  certain  century,  the  chance 
that  it  was  not  distant  more  than  n  years  from  the  middle  of  the  century  is  of 

course  —  ;  hut  if  three  events,  A,  B,  C,  are  known  to  have  occurred  in  the 
ioo 

century,  and  that  A  preceded  B,  and  B  preceded  C,  let  it  he  proposed  to  find 

how  far  this  amount  of  knowledge  alters  the  value  of  the  chance  for  B. 

Let  x  be  the  time  from  the  beginning  of  the  century  to  the  event  B  ;"for 

any  assigned  value  of  x,  the  number  of  triple  cases  is  #(ioo  —  x) :  hence  ^the 

number  of  favourable  cases  divided  by  the  whole  number  is 


JoO+n 
#(ioo  -  x)dx 
50-n 

*  m  Two 3 

l     a;  (ioo  -  x)  dx 
Jo 


--4(-V 

oo         \ioo/ 


2.  Two  numbers,  x,  y,  are  chosen  at  random  between  o  and  a :  find  the 

a2 
chance  that  the  product  xy  shall  be  less  than  —  (its  mean  value). 

4 


Here 


P  = 


\\dxdy 


the  integral  being  limited  by  a  >  x  >  o,  a  >  y  >  o,  and  xy  <  — .    "We  have 

\        « 
accordingly  to  integrate  for  y  from  a  to  o,  when  x  is  between  o  and  - ;  and  from 

4 

—  to  O,  when  x  is  between  -  and  a  ;  thus 
4*  4 

a 

f  *  Ca  a2  a2      a2 

jjdxdy  =      adx  +  1    —  dx  =  —  +  —  log  4. 

JO  la  \X  44 


Hence 


1      I  , 
-+;log». 


3.  Two  points  are  taken  at  random  in  a  given  line  a;  to  find  the  chance 
that  their  distance  asunder  shall  exceed  a  given  value  c. 

It  is  easy  to  see  that  the  distances  of  two  such  points  from  one  end  of  the 
line  are  the  coordinates  of  a  point  taken  at  random 
in  a  square  whose  side  is  a.  Thus  to  every  case 
of  partition  of  the  line  corresponds  a  point  in  the 
square — such  points  being  uniformly  distributed  over 
its  surface. 

Thus,  if  in  the  above  question  x,  y  stand  for  the 
distances  of  the  two  points,  from  one  end  of  the  line, 
y  being  greater  than  x,  we  have  to  find  the  chance 
of  y  —  x  exceeding  c.  The  point  P  whose  co- 
ordinates are  x,  y,  in  the  square  OJD  (side  =  «), 
may  take  all  possible  positions  in  the  triangle  OBD, 
if  no  condition  is  imposed  on  it.  But  if  y  -  x  >  c, 
then  if  we  measure  OS  =  c,  the  favourable  cases  Fig.  62. 


352  On  Mean  Value  and  Probability. 

occur  only  when  P  is  in  the  triangle  BHI ;  hence  the  probability  required 

BHI       /a-cy 
P~  OBJ)~  \a~)  ' 

In  fact  this  is  only  performing  the  integrations  in  the  expression 

["  [""dxdy 
p=^S> . 

("  \Vdxdy 
Jo  Jo 


B 


Z    K 


N 


D 


4.  Two  points  being  taken  at  random  in  a  line  a,  to  find  the  chance  that  no 
one  of  the  three  segments  shall  exceed  a  given 
length  c. 

The  segments  being  as  before,  x,  y  —  x,  a  —  y, 
PH=x,  PK=a-y,  PI  =  y  -  x.  There  will 
be  two  cases  : — 


(1).  Ho -a;  take  OU=B V=DZ=BN=c ; 

then  it  is  easy  to  see  that  the  only  favourable 
cases  are  when  P  falls  in  the  hexagon  UZNMJV; 


P\ 


OBD- 3.  UBZ 
OBD 


(^) 


/ 

) 

V, 

p 

1 

yi 

Fig.  63. 


(2).  If  c  <  -a  ;  take  OJJ  =  BV  =  c,  as  before  ;  then  the  only  favourable 


cases  are  when  P  falls  in  the  triangle  PST; 
BST 


therefore 


p2  = 


OBD 


-(*f)* 


VT  +  RE-  VH 


since  EST '=  -  RT\  and  RT . 

=  2c  —  (a  —  c). 

Such  cases  of  discontinuity  in  the  functions 
expressing  probabilities  frequently  present  them- 
selves.     The  functions  are   connected    by   very 
remarkable  laws.     Thus,  in  the  present  question,     O 
if  Pi  =/{<>)>  P2  =  P(c),  we  have 


f(c)  -/(a  -  c)  =  F(c)  -  F{a  -  c). 


711 


Fig.  64. 


5.  A  floor  is  ruled  with  equidistant  parallel  lines;  a  rod,  shorter  than  the 
distance  between  each  pair,  being  thrown  at  random  on  the  floor,  to  find  the 
chance  of  its  falling  on  one  of  the  lines  (Buffori 's  problem) . 

Let  x  be  the  distance  of  the  centre  of  the  rod  from  the  nearest  line,  6  the 
inclination  of  the  rod  to  a  perpendicular  to  the  parallels,  2a  the  common  distance 
of  the  parallels,  2  c  the  length  of  rod ;  then  as  all  values  of  x  and  6  between  their 


Probabilities.  353 

extreme  limits  are  equally  probable,  the  whole  number  of  cases  will  be  repre- 
sented by 


r 


dxdd 


Now  if  the  rod  crosses  one  of  the  lines  we  must  have  c  >  — —  ;  so  that  the 

CO8  0 


favourable  cases  will  be  measured  by 


tA 


coi0 

dx  =  2C. 


2c 

Thus  the  probability  required  is  p  =  — . 

This  question  is  remarkable  as  having  been  the  first  proposed  on  the  subject 
now  called  Local  Probability.  It  has  been  proposed,  as  a  matter  of  curiosity, 
to  determine  the  value  of  ir  from  this  result,  by  making  a  large  number  of  trials 
with  a  rod  of  length  2a  :  the  difficulty,  however,  here  consists  in  ensuring  that 
the  rod  shall  fall  really  at  random.  The  circumstances  under  which  it  is  thrown 
may  be  more  favourable  to  certain  positions  of  the  rod  than  others.  Though  we 
may  be  unable  to  take  account  a  priori  of  the  causes  of  such  a  tendency,  it  will 
be  found  to  reveal  itself  through  the  medium  of  repeated  trials. 

240.  Sometimes  a  result  depends  upon  a  variable  (or 
variables)  all  the  values  of  which  are  not  equally  probable,  but 
are  such  that  the  probability  of  a  certain  value  for  a  variable 
depends,  according  to  some  law,  on  the  magnitude  of  that 
value  itself  (and  also,  perhaps,  on  the  values  of  other  variables). 
Thus  a  point  may  be  taken  in  a  straight  line  so  that  all 
positions  are  not  equally  probable,  but  the  probability  of  the 
distance  from  one  end  having  the  value  x,  being  proportional 
to  x  itself.  This  would  be  in  fact  supposing  the  series  of 
points  in  question  as  ranged  along  the  line  with  a  density 
proportional  to  x  ;  as,  e.  g.,  if  they  were  the  projections  on  the 
line  of  points  taken  at  random  in  the  space  between  the  line 
and  another  line  drawn  through  one  of  its  extremities.  To 
give  an  example : — 

Two  points  are  taken  in  a  line  a,  with  probabilities 
varying  as  the  distance  from  one  end  A  ;  to  find  the  chance 
of  their  distance  exceeding  a  length  c. 

Let  x,  y,  be  the  distances  from  A,  and  suppose  y  >  x. 


354  On  Mean  Value  and  Probability, 

Here  the  probability  of  a  point  falling  between  x  and  x  +  dx 
is  not  proportional  to  dx,  but  to  xdx  ;  and  the  result  will  be 


xdx 


The  mean  values  of  the  three  divisions  of  the  line,  in  the 
same  case,  will  be  found  to  be 

8  4  i 

—  a.       —  a.       —  a. 
15  i5  i.     5 

The  above  value  of  p  is  also  the  value  of  the  chance,  that 
the  difference  of  the  altitudes  of  two  points  within  a  triangle 

Q 

shall  exceed  a  given  fraction  -  of  the  altitude  of  the  triangle. 

a 


Examples. 

i.  Two  points  being  taken  on  the  sides  OA,  OB,  of  a  square  a2,  the  chance 
of  their  distance  being  less  than  a  given  value  b  is  easily  seen  without  calcula- 
tion to  be  — -,  provided  b  <  a  ;  as  it  is  the  chance  of  a  point  taken  at  random  in 

the  square  falling  within  a  quadrant  of  a  given  circle.  Suppose  now  that  two 
points  are  taken  on  OA,  and  two  on  OB,  and  that  we  take  X,  Y,  the  two  points 
furthest  from  0  on  each  side,  to  find  the  chance  that  their  distance  XYis  less 
than  a  given  length  b  ;  (b  <  a). 

Here  the  probability  of  X  falling  between  *  and  x  +  dx  is  proportional  to 
xdx  ;  likewise  for  y  ;  hence 


P  = 


J  I  xydxdy 
xydxdy 


the  upper  integral  being  limited  by  x2  +  yz  <  bz  ;  hence  p  -  —. 

Thus  it  is  an  even  chance  that  the  point  determined  by  the  coordinates  x,  y 
shall  fall  within  the  quadrant  -  no2. 


Probabilities.  355 

2.  In  a  circular  target  of  area  A  the  area  of  the  bull's  eye  is  «.     If  a 
shot  is  heard  to  strike  the  target,  the  chance  of  its  having  hit  the  bull's  eye  is 

of  course  — -.*     If,  however,  two  shots  have  been  fired,  to  find  the  chance  that 
A 

the  best  of  tbe  two  has  hit  the  bull's  eye. 

This  is  easily  solved  by  elementary  considerations  ;  as  the  chance  of  both 
missing  the  bull's  eye  is 

(A- ay 

Hence  the  required  chance  of  the  best  shot  having  hit  it  is 


a  (         a  \ 

-^a[2-a) 


3.  Let  it  be  proposed,  however,  to  find  the  chance  of  the  best  of  the  two 
shots  (i.  e.  that  nearest  the  centre)  having  hit  any  given  area  a,  traced  out  on 
the  target. 

The  number  of  cases  in  which  the  worst  shot  falls  on  any  element  dS,  at  a 
distance  r  from  the  centre,  is  measured  by  irr^dS ;  hence  the  chance  of  the  worst 
shot  striking  the  area  a  is 

_$fr*dS  (over  a)  __  m 
P~)$r*dS{oYeTA)~  M' 

where  M,  m  are  the  moments  of  inertia  of  A,  a  round  the  centre  of  the  target. 
Now,  the  probability  of  both  shots  missing  a  is 

A  -a\2 


£r)' 


hence  that  of  a  being  hit  (by  one  or  both)  is 


m> 


and  the  chance  of  both  hitting  it  is  — .     But  the  chance  of  a  being  hit  is 

A 

chance  of  best  +  chance  of  worst  —  chance  of  both  ; 

hence  if  p\  be  the  required  chance,  viz.,  of  the  best  shot  striking  a, 


m       a-  I  A.  —  a\ '  am 


where  m,  M  are  the  moments  of  inertia  above. 

Or,  we  might  have  considered  the  number  of  cases  in  which  the  best  shot 
falls  on  the  element  dS,  viz.,  tt(.#2  -  r2)dS,  where  It  =  radius  of  target.  This 
would  have  given  the  required  probability 

Rla  —  m 


M2A-M' 
which  is  easily  shown  to  be  identical  with  the  above  value. 

_  *  That  is,  disregarding  the  effect  of  the  aim  directing  it  with  greater  proba- 
bility to  the  centre  of  the  target.  This  would  be  practically  correct  in  the  case 
of  a  very  bad  marksman,  who  frequently  misses  the  target  altogether. 


356 


On  Mean  Value  and  Probability. 


241.  Carve  of  Frequency. — In  questions  relating  to 
a  variable  the  probability  of  any  value  of  which  is  a  function 
of  that  value  itself,  it  is  often 
useful  to  consider  what  is  called 
a  curve  of  frequency.  Thus,  if 
the  probability  of  a  given  value 
of  x  is  proportional  to  0  (x),  and 
we  draw  a  curve  y  -  C #(#), 
then  when  a  great  number 
of  values  for  x  are  taken,  the 
number  in  any  element  dx  is 
proportional  to  the  area  of  the  curve  standing  on  that 
element  ;  the  ordinate  at  any  point  P  representing  the 
density  or  frequency  of  the  points  at  P  :  the  abscissas  of  all 
points  taken  at  random  in  the  area  of  the  curve  are  equally 
probable. 

Thus,  if  two  points  X,  Fare  taken  at  random  in  a  straight 

line  AB,  and  X  means  always  that  nearest  to  A,  the  curve 

of  frequency  for  T  will  be  a  straight  line  through  A ;  that 

for  X  a  straight  line  through  B.     This  will  often  simplify 

questions :  e.g.  suppose  we  have  to  find  what  is  sometimes 

called  the  most  probable  value  for  A  Y,  i.  e.  suoh  a  value 

AP  that  A  Y  is  equally  likely  to  exceed  or  to  fall  short  of  it. 

Since   the    curve    of    frequency   for 

Y  is  a  line  AC,  we  have  only  to 

find   P,    so    that    PD    bisects    the 

AB 
triangle    ABC  ;     i.  e.     AP  =  — — 

because  as  many  values  of  AY 
exceed  AP  as  fall  short  of  it. 
The    most    probable    value    is   not 

2 
the  mean  value,  viz.,  -  AB,  being  the  horizontal  distance  of 

the  centre  of  gravity  of  ABC,  from  A. 

A  point  Y  is  taken  at  random  in  a  line  AB  =  a,  and 
then  a  point  X  is  taken  at  random  in  A  Y  (or  a  rod  may  be 
supposed  broken  in  two  at  random,  and  one  of  the  pieces 
then  broken  in  two),  to  find  the  chance  of  the  length  of  AX 
falling  within  given  limits. 

Let  x,  y,  be  the  distances  from  A ;  for  any  assigned  value 


Curve  of  Frequency. 


357 


of  y,  the  chance  of  X  falling  between  x  and  x  +  dx  is  —  ; 

hence  the  chance  of  X  falling  between 
x  and  x  +  dx,  and  of  Y  falling  between 
y  and  y  +  dy,  is  measured  by 

dxdy 
ay    ' 

hence  the  whole  chance  of  X  falling 
between  x  and  x  +  dx  is 

dx 


lx[a  dy      dx ,      a  _   . 

—     —  =  — 1°£  —  -  —  dx  log  a?, 


if  for  simplicity  we  put  a  =  i . 

Thus  the  curve  of  frequency  for  X  is  a  logarithmic  curve 
BR,  whose  ordinate  is 

2  =  -l0g#, 

the  frequency  at  A  being  infinitely  great. 
The  area  of  this  curve  from  o  to  x  is 

xlog6-; 

X 

and  this  is  the  probability  of  AX  being  between  o  and  x ; 
the  whole  areaa  when  x  =  i,  being  i,  as  it  ought  to  be,  as 
it  is  certain  that  X  falls  in  AB.  The  chance  of  X  falling 
between  given  limits  x',  x"  is  of  course 


x'log-,-x"log-p. 

X  X 

To  find  the  most  probable  value  of  x  we  should  have  to 
solve  the  equation 

x(i  -  logx)  =  -. 


This  gives  x  about  one-fifth  of  the  line  AB. 


358  On  Mean  Value  and  Probability. 

The  mean  value  of  x  is 

xzdx 

M  mk one-fourth  of  AB. 

zdx 

This  last  result  might  have  been  foreseen :  because  if  we 
take  a  point  at  random  in  each  of  the  segments  AY,  YB, 
the  line  AB  is  divided  into  four  parts,  the  mean  values  of 
which  must  be  the  same,  as  each  of  them  goes  through  the 
same  series  of  values  as  the  others;  the  sum  of  the  mean 
values  being  AB. 

Examples. 

i.  A  line  is  divided  at  random,  and  one  of  the  parts  again  divided  at  random 
as  above,  to  find  the  chance  that  no  one  of  the  three  parts  shall  exceed  the  sum 
of  the  other  two  (i.e.  that  a  triangle  might  be  formed  by  them).  {Cambridge 
Math.  Tripos,  1854.) 

The  probability  that  X,  Y  shall  be  taken  in  two  assigned  elements  dz,  dy 
is  (taking  a  -  i), 

dxdy 


This  differential  being  integrated  throughout  any  limits  gives  the  sum  of  the 
probabilities  of  X,  Y  being  found  in  each  pair  of  values  for  dz  and  dy  which 
enter  into  the  summation: — that  is,  the  cases  being  mutually  exclusive,  the 
probability  that  X,  Y  will  be  found  in  some  one  of  those  pairs. 
In  the  present  case  the  limits  are  equivalent  to 

1  1 

z<-<y  <  r,     z>y  --. 

_                                                 f1  fi    dydz      ,  I 

Hence  P  =\  =  log  2  — . 

h  h-i    y  2 

2.  An  urn  contains  a  large  number  of  black  and  white  balls,  the  proportion 
of  each  being  unknown :  if  on  drawing  tn  +  n  balls,  tn  are  found  white  and 
n  black,  to  find  the  probability  that  the  ratio  of  the  numbers  of  each  colour  lies 
between  given  limits. 

The  question  will  not  be  altered  if 
we  suppose  all  the  balls  ranged  in  a  line 
AB,  the  white  ones  on  the  left,  the 
black  on  the  right,  the  point  X  where 
they  meet  being  unknown,  and  all  posi- 
tions for  it  in  AB  being  a  priori  equally  ?~ =t it- 
probable  ;  then  tn  +  n  points  being  taken,  "p-  ar 
at  random  in  AB,  m  are  found  to  fall  on  8"  "°- 
AX,  n  on  XB.    That  is,  all  we  know  of  X  is,  that  it  is  the  (w  +  i)'*  in  order, 


Curve  of  Frequency.  359 

beginning  from  A,  of  m  +  n  +  i  points  falling  at  random  in  AB.     If  AX  =  x, 
AB  =  i,  the  number  of  cases  for  X  between  x  and  x  +  dx  is  measured  by 

I  m  +  n 

L  xm  (i  -  x)ndx.* 

Hence  tbe  probability  that  the  ratio  of  the  white  balls  in  the  urn  to  the 
whole  number  lies  between  any  two  given  limits  a,  £ — that  is,  that  the  distance 
from  A  of  the  point  X  lies  between  o  and  >3 — is 


xm(i  -x)ndx 


1    xm(] 
Jo 


;i  -%ydx 

The  curve  of  frequency  for  the  point  X  will  be  one  whose  ordinate  is 
y  =  xm  (i  -  x)n. 

The  maximum  ordinate  KV  occurs  at  a  point  K,  dividing  AB  in  the  ratio 
m  :  n.  This  is  of  course  what  we  should  expect :  the  ratio  of  the  numbers  of 
black  and  white  balls  is  more  likely  to  be  that  of  the  numbers  drawn  of  each 
than  any  other.  The  value  for  p  above  is  simply  the  area  of  the  above  curve 
between  the  values  a,  fi,  of  x,  divided  by  the  whole  area. 

Let  us  suppose,  for  instance,  that  3  white  and  2  black  balls  have  been 

drawn  ;  to  find  the  chance  that  the  proportion  of  white  balls  is  between  -and  - 

1  3 

of  the  whole— that  is,  that  it  differs  by  less  than  ±  -  from  -,  its  most  natural 

value. 

(J  *<*-**     2256      ,8  , 

p  =  J-| ^^^  nearly. 

The  above  results  will  apply  to  any  event  that  must  turn  out  in  one  of 
two  ways  which  are  mutually  exclusive,  this  being  the  whole  of  our  d  priori 
knowledge  with  regard  to  it — the  ratio  of  the  black,  or  wbite  balls  to  the 
whole  number,  meaning  the  real  probability  of  either  event,  as  would  be 
manifested  by  an  infinite  number  of  trials.  We  will  give  one  more  example  of 
the  same  kind. 

3.  An  event  has  happened  m  times  and  failed  n  times  in  m  +  n  trials.  To 
find  the  probability  that,  on  p  +  q  further  trials,  it  shall  happen  p  times  and 
failf^  times. 

*  For  a  specified  set  of  m  points,  out  of  the  m  +  n,  falling  in  AX}  the 

\m  +  n 

number  is  xm  (1  -  x)ndx  ;  the  number  of  such  sets  is  -7 — : — . 
v  '  [  m  f  n 


360 


On  Mean  Value  and  Probability. 


That  is,  that  p  +  q  more  points  being  taken  at  random  in  AB,  p  shall  fall  in 
AX,  and  q  in  BX.     The  whole  number  of  cases  is  as  before 


[m  4  n 


*♦»  (i  -  x)»dx 


zm(i  -#)'»<&;. 


When  any  particular  set  of  j?  points,  out  of  the  p  +  q  additional  trials,  falls  in 
AX,  the  number  of  favourable  cases  is 


(i  -x)n+*dx. 
But  the  number  of  different  sets  of  p  points  is ■ 


(p  +  q) 


Hence  the  probability  is,  putting  a#  before  I  p  for 

fp  +  q    ^x^p{ir-x)^dx 


3  .  .  .  p  .  I  .  2 
2  •  3  •  •  •  P, 


Pi 


[i -It 


\    :r"»(l 

Jo 


x)ndx 


Pi 


By  means  of  the  known  values  of  these  definite  integrals  (p.  117),  we  find 
\p  +  q     [m+p  [n  +  q        \m  +  n  +  I 
[^[^   '         [m[n  [m  rn  +  p  +  q+l' 

For  instance,  the  chance  [that  in  one  further  trial  the  event  shall  happen  is 
This  is  easily  verified,  as  the  line  AB  has  been  divided  into  m  +  n  +  2 


m  +  n  +  2 

sections  by  the  m  +  n  +  1  points  in  it,  including  X.  Now,  if  one  more  trial  is 
made,  i.  e.  one  more  point  taken  at  random,  it  is  equally  likely  to  fall  in  any 
section  ;  and  m  +  I-  sections  out  of  the  entire  number  are  favourable. 

4.  Trace  the  curve  of  frequency  of  the  ratio  T ;  a  and  b  being  numbers  taken 

0 
at  random  within  the  limits  +  I. 

If  we  measure  the  values  of 
the  ratio  as  abscissas  along  an 
axis  OX,  and  make  OA  =  1, 
OA'  =  -  I,  AB=  A'B'  =  I  ; 
then  the  line  whose  ordinates  C 
are  proportional  to  the  fre- 
quency will  be,  for  values  of 

a 

-  comprised  between  the  limits 

0 

+  1,  the  straight  line  BB' ;  but,  for  values  beyond  these  limits,  will  consist  of 

the  arcs  BC,  B'C  of  the  curve  x2y  =  1. 

a 
It  is  thus  an  even  chance  that  the  ratio  -  lies  itself  between  the  limits  ±  1  : 

0 

this  would  also  appear  by  a  construction  such  as  that  given  in  the  next  Article. 


Fig  69. 


Errors  of  Observation. 


361 


242.  Errors  of  Observation. — One  of  the  most  im- 
portant, practically,  as  well  as  the  most  difficult,  departments 
of  the  theory  of  Prohahility  is  that  which  treats  of  Errors  of 
Observation.  We  will  give  here  an  example  of  the  simplest 
description. 

Two  magnitudes  A  and  B  are  measured  ;  each  measure- 
ment being  subject  to  an  error,  of  excess  or  defect,  which 
may  amount  to  ±  a,  all  values  between  these  limits  being 
supposed  equally  probable.*  To  determine  the  probability 
that  the  error  in  the  sum,  A  +  B,  of  the  two  magnitudes, 
shall  lie  within  given  limits ;  also  its  mean  value. 

Thus  the  horizontal  angular  distance  of  two  objects  A,  C 
is  sometimes  found  by  measuring  the  angle  between  A  and  B, 
an  intermediate  object ;  and  afterwards  that  between  B  and 
(7,  and  adding  the  two  angles.  If  each  measurement  is  liable 
to  an  error  ±  5',  all  values  being  equally  probable,  to  find  the 
probability  of  the  error  of  the  result  falling  within  assigned 
limits  :  its  extreme  limits  being  of  course  ±  1  o'. 

The  question  is  more  easily  comprehended  by  means  of 
a   geometrical   construction   than   by  B'  K 

integration. 

Take  AB  =  2a ;  then  all  the  values 
of  the  first  error  are  the  distances 
from  0  of  points  P  taken  at  random 
in  AB  ;  positive  when  in  OB ; 
negative  when  in  OA.  Make  also 
A'B'  =  2a;  the  values  of  the  second 
error  are  given  by  points  in  A!B\ 
Take  any  values,  OP  =  x  for  the  first, 
OP'  =  x  for  the  second :  these  values 
taken  as  co-ordinates  determine  a  point  V  corresponding  to 
one  case  of  the  compound  error  x  +  x' ;  and  such  points  V 
will  be  uniformly  distributed  over  the  square  HK.  The  value 
of  the  compound  error  c  corresponding  to  the  point  V  is 

£  =  x  +  x'  =  OS, 

if  VS  be  drawn  at  450  to  the  axes.     Now  all  values  of  the 


p' 

V, 

0 

1 

A 

Fig.  70. 


*  This  supposition  must  not  be  taken  to  be  practically  correct.  The  Theory 
of  Errors  shows  that  the  probability  of  an  error  of  magnitude  x  is  proportional 
to  e~<»\ 

[24] 


362  On  Mean  Value  and  Probability. 

errors  x,  xf  which  give  x  +  x  the  same,  give  the  same  value 
for  t ;  hence  all  points  on  the  line  JI  correspond  to  com- 
pound errors  of  amount  OS.  Take  Ss  =  de ;  the  number  of 
compound  errors  between  e  and  e  +  de  is  the  number  of 
points  between  JI  and  a  parallel  to  it  through  s.  Now  the 
area  of  this  infinitesimal  strip  is  evidently 

(2a  -  e)de. 

Hence  the  probability  of  the  error  being  between  e  and 
€  +  de  is 

(2a  —  e)  de 

4a 

This  holds  for  negative  values  of  e,  provided  we  only  oonsider 
their  arithmetical  magnitude. 

Thus  the  frequency  of  an  error  of  magnitude  e  =  OS  is 
proportional  to  JI,  the  intercept  of  a  line  through  S  sloping 
at  450.  The  probability  of  the  error  e  falling  between  any 
two  given  limits  OS,  OS'  is  found  by  measuring  these 
lengths  (with  their  proper  signs)  from  0,  along  AB,  and 
dividing  the  area  intercepted  on  the  square  by  parallels 
through  S,  S'  sloping  at  450,  by  4a2,  the  area  of  the  whole 
square. 

Thus  the  chance  of  the  error  falling  between  the  limits 

±  a  (those  of  the  two  component  errors)  is  -. 

The  mean  value  of  the  error,  strictly  speaking,  is  o ;  but  it 
is  evident  that  for  this  purpose  we  ought  to  consider  negative 
errors  as  positive ;  and  consequently  take  the  mean  of  the 
arithmetical  values  of  all  the  errors,  whioh  is  the  same  as  the 
mean  of  the  positive  errors  only ;  hence  the  mean  error 
required  is 

M(e)=±-a. 
3 

The  most  probable  value,  such  that  it  is  an  even  chance  that 
the  error  exceeds  it  (since  the  triangle  JKI  must  be  -  of  the 
whole  square,  for  that  value  of  OS),  is 

±  a  (2  -  0/2)  =  ±  .586  a. 


Errors  of  Observation.  363 

Let  it  be  now  proposed  to  find  the  probability  of  a  given 
error  in  the  sum  of  A  and  i?,  assuming,  according  to  the 
modern  theory  of  errors,  that  the  probability  of  an  error  be- 
tween x  and  x  +  dx  in  either  is 


dx; 


V'i 


the  coefficient  — —  being  determined  by  the  necessary  con- 

dition  that  the  differential,  being  integrated  from  oo  to  -  oo, 
must  give  unity  ;  as  the  error  must  lie  between  these  limits.* 
Referring  to  the  above  construction,  the  number  of  values 
of  the  first  error  between  x  and  x  +  dx  being  proportional  to 


c~c°dx, 

and  the  number  of  values  of  the  second  between  x  and  x'  +  dx 
proportional  to 

X* 

e~~*dx, 

the  corresponding  number  of  values  of  the  compound  error  is 
proportional  to 

x*  +x'<> 

e     c*    dxdx' . 

Hence  the  number  of  points,  corresponding  each  to  a  case 
of  the  compound  error,  in  any  element  dS  of  the  plane  at  a 
distance  r  from  the  origin,  is  measured  by 


e~ca~dS; 
which  shows  that  the  points  have  the  same  density  along  any 


*  It  is  of  course  absurd  to  consider  infinite  values  for  an  error :  but  the 

X* 

curve  y  =  e  c2  tends  so  rapidly  to  coincide  with  its  asymptote,  the  axis  of  x, 
that  the  cases  where  x  has  any  large  values  are  so  trifling  in  number,  that  it  is 
indifferent  whether  we  include  them  or  not. 

[34  a] 


364  On  Mean  Value  and  Probability. 

circle  whose  centre  is  0.  Now  the  probability  of  this  com- 
pound error  being  between  c  and  e  +  dt  is  proportional  to  the 
number  of  points  between  J  I  and  the  consecutive  line;  making, 
as  before,  08  =  c,  Ss  =  de.  But  this  number  is  the  same 
as  when  the  strip  JI  is  turned  round  0  through  an  angle  of 
45°,  because  the  points  lie  in  concentric  circles  of  equal  den- 
sity.    Hence  the  number  is  proportional  to 

e-^dJ_X  ccadx  =  C-^e~^de, 

as  the  perpendicular  from  0  on  JI  is  — =.. 

Thus  the  probability  of  a  compound  error  between  e  and 
e  +  de  is  proportional  to 

e       da; 

and  as  this,  when  integrated  between  the  limits  ±  oo ,  must 
give  the  probability  i,  the  value  of  p  is 

i        .H 
p  = —  e  2c*  de. 

It  thus  follows  the  same  law  as  the  two  component  errors, 
c  v/2  taking  the  place  of  c. 

243.  Various  artifices  have  been  employed  for  the  solution 
of  different  interesting  questions  on  Probability,  which  would 
be  found  extremely  tedious,  or  impracticable,  if  attempted 
by  direct  integration.     For  example : 

Two  points  are  taken  at  random  within  a  sphere  of  radius 
r ;  to  find  the  chance  that  their  distance  is 
less  than  a  given  value  c. 

Let  F  =  number  of  favourable  cases, 
W  =  whole  number ;  then 

Let  us  consider  the  differential  dF9  or  Fis-  71- 

the  additional  favourable  cases  introduced  by  giving  r  the 
increment  dr,  c  remaining  unchanged. 


Errors  of  Observation.  365 

If  one  of  the  points  A  is  taken  anywhere  (at  P)  in  the 
infinitesimal  shell  between  the  two  spheres,  then  drawing  a 
sphere  with  centre  P,  radius  c,  all  positions  of  the  second 
point,  P,  in  the  lens  ED  common  to  the  two  spheres,  are 
favourable ;  let  L  =  volume  ED,  then  the  number  of  favour- 
able cases  when  A  is  in  the  shell  is 

4irr2dr.L: 

doubling  this,  for  the  cases  when  B  is  in  the  same  shell, 

dF=8irrLdr. 

Now  it  may  be  easily  proved,  from  the  value  for  the  volume 
of  a  segment  of  a  sphere,  that 

T  27T    ,         7TC4 

L  =  —  c3 ; 

3  $r 

hence  F  =  8tt2  (-  cz  r3  -  i  c4  r2  +  C 

C  being  an  unknown  constant ;  i.e.  involving  c,  but  not  r ; 

.,    '                                 F         &       q    c4  9  C 

therefore            p  =  -r =  -  -  ~ .  -  +  -  — . 

r      16             r3      16   r4  2  r6 

—  7rV 
9 

Now  the  probability  =  i  if  r  =  -  c ; 

therefore     i  =  8  -  o  +     x  64  -  ;     .*.    -C=  —  c6: 
2  c6  2        64 

C3  Q    C4  I    C^ 

therefore  0  = — —  + . 

r3      i6r4      32  r6 

If  the  two  points  be  taken  within  a  circle,  instead  of  a 
sphere,  it  may  be  proved  by  a  similar  process  that 

c2      2  f       c2\   .        c        I     c  f       c2\  I       ? 
r"      7r\       r2/  2r      4?r   r\       r2J\       r2 


366  On  Mean  Value  and  Probability. 

It  is  a  very  remarkable  fact,  pointed  out  by  Mr.  S.  Roberts, 
that  if  we  draw  the  chord  ED,  the  probability  is,  in  the  case 
of  the  circle, 

_  2  .  segment  EQD  +  segment  EPD 
area  of  circle  EHD 

and  also,  in  the  case  of  the  sphere, 

2  .  volume  EQD  +  volume  EPD 


P  = 


volume  of  sphere  EHD 


These  results  evidently  suggest  that  there  must  be  some 
manner  of  viewing  the  question  which  would  conduct"  to 
them  in  a  direct  way. 

Examples. 

i.  Three  points  being  taken  at  random  within  a  sphere,  to  find  the  chance 
that  the  triangle  which  they  determine  shall  be  acute-angled. 

As  the  probability  is  independent  of  the  radius  of  the  sphere,  it  is  easy  to 
see  that  we  may  take  the  farthest  from  the  centre  of  the  three  points  as  fixed  on 
the  surface  of  the  sphere.  For  if  p  be  the  probability  of  an  acute-angled  triangle 
in  this  case,  p  will  also  be  the  probability  of  an  acute-angled  triangle  for  each 
position  of  the  farthest  point,  as  it  travels  over  the  whole  volume  of  the  sphere. 
Hence  p  will  be  the  probability  when  no  restriction  is  put  on  any  of  the  points. 

Take  then  A,  one  of  the  points  on  the  surface  of  the  sphere ;  two  others,  B,  C, 
being  taken  at  random  within  it,  and  let  us  find  the 
chance  of  ABC  being  obtuse-angled :  to  do  this,  we 
will  find  separately  the  chance  of  the  angles  A,  B,  G 
being  obtuse :  the  events  being  mutually  exclusive, 
the  probability  required  will  be  the  sum  of  these 
three. 

(i).  To  find  the  chance  that  A  is  obtuse,  let  us  fix 
B  ;  then,  drawing  the  plane  A  V  perpendicular  to  AB, 
the  chance  required  is 

volume  of  segment  AST 
volume  of  sphere 

Let  r  m  OA,  the  radius  of  sphere,  p  =  AB,  6  =  L  OAB  ;  then  the  volume  of 
the  segment  AHV  is 

£nr3(i  -  cos  Bf  (2  +  cosfl); 
therefore  when  B  is  fixed  the  ohance  is 

£(1  -COS6)2  (2  +  CO3  0). 


Errors  of  Observation.  367 

Now  let  B  move  oyer  the  whole  volume  of  the  sphere,  and  we  have  for  the 
probability  Pa,  that  A  is  obtuse 


J*2r2rcos,0 
(2  -  3  cos  0  +  oos3  0)  p2  sin  0 ^0  dp. 


8r> 

Hence  P^=— . 

70 

(2).  To  find  the  chance,  PB,  that  B  is  obtuse.  Fix  B  as  before  ;  then  the 
chance  that  B  is  acute  is 

segment  MEN 
sphere 

Now,  volume  MEN  =  \irrz  (  -  +  1  -  cos  0  J    (  2  +  cos  0  -  -  J ;  so  that  the 

chance  is 

if  p  p2  p3 

-  <  2  -  3  cos  0  +  cos30  +  3  -  (1  -  cos20)  +  3  r—  COS  0  -  h: 
4  (  r  r*  r3 

Hence  the  whole  probability  ( 1  -  PB)  that  B  is  acute  is 

jr 
^     f2p2rcos0/  p  p2  p3  \ 

—  \  ja-3cos0  +  cos30+3-(i  -cos20)  +  3^cos0-r3j  p2  sm  6  d8  dp. 

Performing  the  integrations,  we  find  PB  =  —  • 

The  probability  for  G  is,  of  course,  the  same  as  for  B  ;  hence  the  whole  pro- 
bability of  an  obtuse-angled  triangle  is 

P=Pa  +  Pb  +  Pc=  —  +  ^  +  — =  — . 
70      70      70      70 

Hence,  the  chance  of  an  acute-angled  triangle  is  — . 

70 

For  three  points  within  a  circle  the  chance  of  an  acute-angled  triangle  is 
±      X 

IT2        8* 

2.  Two  points,  A,  B,  are  taken  at  random  in  a  triangle.  If  two  other  points, 
C,  D,  are  also  taken  at  random  in  the  triangle,  find  the  chance  that  they  shall  lie 
on  opposite  sides  of  the  line  AB. 


368  On  Mean  Value  and  Probability. 

The  sides  of  the  triangle  ABC  produced  divide  the  whole  triangle  into  seven 
spaces.  Of  these,  the  mean  value  of 
those  marked  (a)  is  the  same,  viz.,  the 
mean  value  of  ABC;  or,  ■&  of  the 
whole  triangle,  as  we  have  shown  in 
Art.  236 ;   the  mean  value   of  those  P 

marked  (IS)  being  $  of  the  triangle. 

This  is  easily  seen :  for  instance, 
if  the  whole  area  =  1,  the  mean  value 
of  the  space  PBQ  gives  the  chance 
that  if  the  fourth  point  D  be  taken 
at  random,  B  shall  fall  within  the 
triangle  ABC:  now  the  mean  value 
of  ABC  gives  the  chance  that  D  shall  /_ 
fall  within  ABC ;  but  these  two 
chances  are  equal.  Fig.  73. 

Hence  we  see  that  if  A,  B,  C  be 
taken  at  random,  the  mean  value  of  that  portion  of  the  whole  triangle  which 
lies  on  the  same  side  of  AB  as  C  does  is  H  °f  the  whole ;  that  of  the  opposite 
portion  is  tV 

Henoe  the  chance  of  Cand  JD  falling  on  opposite  sides  of  AB  is  tV- 

244.  Random  Straight  Lines. — If  an  infinite  number 
of  straight  lines  be  drawn  at  random  in  a  plane,  there  will 
be  as  many  parallel  to  any  given  direction  as  to  any  other, 
all  directions  being  equally  probable  ;  also  those  having  any 
given  direction  will  be  disposed  with  equal  frequency  all 
over  the  plane.  Hence,  if  a  line  be  determined  by  the  co- 
ordinates p,  w,  the  perpendicular  on  it  from  a  fixed  origin  0, 
and  the  inclination  of  that  perpendicular  to  a  fixed  axis ;  then 
if  p,  u>  be  made  to  vary  by  equal  infinitesimal  increments, 
the  series  of  lines  so  given  will  represent  the  entire  series  of 
random  straight  lines.  Thus  the  number  of  lines  for  which 
p  falls  between  p  and  p  +  dp,  and  u>  between  w  and  w  +  dw, 
will  be  measured  by  dpdio,  and  the  integral 

jj  dpdcjy 

between  any  limits,  measures  the  number  of  lines  within  those 
limits. 

It  is  easy  to  show  from  this  that  the  number  of  random 
lines  which  meet  any  closed  convex  contour  of  length  L  is 
measured  by  L. 

For,  taking  0  inside  the  contour,  and  integrating  first 
for  p,  from  o  to  jo,  the  perpendicular  on  the  tangent  to  the 
contour,  we  have  jpdw  :  taking  this  through  four  right  angles 


Random  Straight  Lines. 


369 


for  w,  we  have  by  Legendre's  theorem  (p.  232),  N  being  the 
measure  of  the  number  of  lines, 


N 


pdb 


Thus  if  a  random  line  meet  a  given  contour,  of  length  L, 
the  chance  of  its  meeting  another  convex  contour,  of  length  /, 
internal  to  the  former,  is 

I 

If  the  given  contour  be  not  convex,  or  not  closed,  iV  will 
evidently  be  the  length  of  an  endless  string,  drawn  tight 
around  the  contour. 

Examples. 

1.  If  a  random  line  meet  a  closed  convex  contour,  of  length  X,  the  chance 
of  it  meeting  another  such  contour,  external  to  the  former,  is 


P  = 


where  X  is  the  length  of  an  endless  band 
enveloping  both  contours,  and  crossing 
between  them,  and  Y  that  of  a  band  also 
enveloping  both,  but  not  crossing. 

This  may  be  shown  by  means  of 
Legendre's  integral  above ;  or  as  fol- 
lows : — 

Call,  for  shortness,  N{A)  the  number 
of  lines  meeting  an  area  A ;  N(A,  A') 
the  mimber  which  meet  both  A  and  A' : 


then 


N{SROQPH)  +  N(S'Q'OR'P'H')  =  N(SROQPH  +  S'QOR'P'H') 

+  N(SROQPH,  S'Q'OR'P'B'), 

since  in  the  first  member  each  line  meeting  both  areas  is  counted  twice.  But 
the  number  of  lines  meeting  the  non-convex  figure  consisting  of  OQPHSR  and 
OQ'S'H'P'R'  is  equal  to  the  band  Y,  and  the  number  meeting  both  these  areas 
is  identical  with  that  of  those  meeting  the  given  areas  Q.,  Df;  hence 

x=  r+iV(n,  a'). 


Thus  the  number  meeting  both  the  given  areas  is  measured  by  X  -  Y.     Hence 
the  theorem  follows. 


370 


On  Mean  Value  and  Probability. 


2.  Two  random  chords  cross  a  given  convex  boundary,  of  length  Z,  and  area 
A ;  to  find  the  chance  that  their  intersection  falls  inside  the  boundary. 

Consider  the  first  chord  in  any  position :  let  C  be  its  length  ;  considering  it 
as  a  closed  area,  the  chance  of  the  second  chord  meeting  it  is 


2C 

L  '' 


and  the  whole  chance  of  its  co-ordinates  falling  in 
chord  meeting  it  in  that  position,  is 


da,  and  of  the  second 


2(7  dp  da 


■z^Cdp  da. 


L  jjdpda      Z2 
But  the  whole  chance  is  the  sum  of  these  chances  for  all  its  positions 


therefore 


prob.  =  —  I  I  Cdpda. 


Now,  for  a  given  value  of  a,  the  value  of  J  Cdp  is  evidently  the  area  A  ;  then 
taking  a  from  ir  to  o, 

required  probability  =  -=-. 

The  mean  value  of  a  chord  drawn  at  random  across  the  boundary  is 
J  j  Cdpda      irCl 
Hdpda        Z* 

3.  A  straight  band  of  breadth  c  being  traced  on  a  floor,  and  a  circle  of  radius 
r  thrown  on  it  at  random,  to  find  the  mean  area  of  the  band  which  is  covered  by 
the  circle.     (The  cases  are  omitted  where  the  circle  falls  outside  the  band.)* 

If  S  be  the  space  covered,  the  chance  of  a  random  point  on  the  circle  falling 
on  the  band  is 

M(8) 


This  is  the  same  as  if  the  circle  were  fixed,  and  the  band  thrown  on  it  at 
random.      Now  let  A  be  a  position  of  the 

random  point :  the  favourable  cases  are  when  ,--' ~~-^X^ 

HK,  the  bisector  of  the  band,  meets  a  circle,  /  s' 

centre  A,  radius  \  c ;  and  the  whole  number 
are  when  HK  meets  a  circle,  centre  0,  radius 
r  +  \c ;  hence  (Art.  236)  the  probability  is 


P  = 


27T.  JfC 
2ir(r  +  %c) 


c 
2r+c 


This  is  constant  for  all  positions  of  A  ; 
hence,  equating  these  two  values  of  p,  the 


Fig.  75- 


*  Or  the  floor  may  be  supposed  painted  with  parallel  bands,  at  a  distanoe 
asunder  equal  to  the  diameter  ;  so  that  the  circle  must  fall  on  one. 


Application  to  Evaluation  of  Definite  Integrals.        371 

mean  area  required  is 

M(S)  =  — —  tit2. 
v  2r  +  c 

The  mean  value  of  the  part  of  the  circumference  which  falls  on  the  hand  is 

the  same  fraction of  the  whole  circumference. 

2r  +  c 
If  any  convex  area  Cl,  of  perimeter  Z,  be  thrown  on  the  band,  instead  of  a 
circle,  the  mean  area  covered  is 

ire 

M{S)  = n. 

v    '      L  +  ire 

245.  Application  to  Evaluation  of  Definite  Inte- 
grals.— The  consideration  of  probability  sometimes  may  be 
applied  to  determine  the  values  of  Definite  Integrals.  For 
instance,  if  n  +  1  points  are  taken  at  random  in  a  line,  /,  and 
we  consider  the  chance  that  one  of  them,  X,  shall  be  the  last, 
beginning  from  the  end  A  of  the  line,  the  number  of  favour- 
able cased,  when  X  is  the  element  dx,  is,  calling  AX,  x, 

xn  dx. 
Henoe 


1: 


xndx 
Jo 

p 


but  the  chance  must  be :  we  thus  have  an  independent 

n  +  1  A 

proof  that 

fl                       in  +  i 
x»dx  = , 
»  +  I 

when  n  is  an  integer. 

Again,  if  m  +  n  +  1  points  are  taken,  to  find  the  chance 
that  X  shall  be  the  (m  +  i)thiii  order ;  the  number  of  favour- 
able cases,  when  X  falls  in  dx,  and  a  particular  set  of  m  points 
falls  to  the  left  of  X,  is 

irm(i  -  x)  ndx ;  taking  /  =  1  ; 
hence  the  whole  number  of  favourable  cases  is 


==\    %m(i  -  x)ndx; 
m\_nj0 


372  On  Mean  Value  and  Probability. 

this  is  the  required  probability,  since  /m+n+1  =  i.     But  the 

value  is  ,  as  every  point  is  equally  likely  to  fall  in 

the  (m  +  i)th  place  :  we  thus  deduce  the  definite  integral 

|  m  [n 


xm(i  -x)ndx  = 


m  +  n  +  i 


when  mf  n  are  integers.     (See  Art.  92.) 

246.  To  investigate  the  probability  that  the  inclination 
of  the  line  joining  any  two  points  in  a 
given  convex  area  £2  shall  lie  within 
given  limits. 

We  give  here  a  method  of  reducing 
this  question  to  calculation,  for  the  sake 
of  an  integral  to  which  it  leads,   and  QJ 
which  is  not  easy  to  deduce  otherwise. 

First,  let  one  of  the  points,  A,  be 
fixed  ;  draw  through  it  a  chord  PQ  =  C,  ^ig  ?6 

at  an  inclination  9  to  some  fixed  line ; 
put  AP  =  r,  AQ  =  r' ;  then  the  number  of  cases  for  which 
the  direction  of  the  line  joining  A  and  B  lies  between  9  and 
0  +  dO  is  measured  by 

l(r2  +  r'2)dO. 

Now,  let  A  range  over  the  space  between  PQ  and  a 
parallel  chord  distant  dp  from  it,  the  number  of  cases  for 
which  A  lies  in  this  space,  and  the  direction  of  AH 
from  9  to  9  +  d9,  is  (first  considering  A  to  lie  in  the 
element  drdp) 


idpd9i    (r 


+  r'2)dr  =  iC3dpd9. 


Let  p  be  the  perpendicular  on  C  from  a  given  origin  O, 
and  let  to  be  the  inclination  of  p  (we  may  put  dw  for  d9),  then 
C  will  be  a  given  function  of  p,  to  ;  aad  integrating  first  for  w 
constant,  the  whole  number  of  cases  for  which  w  falls  between 
given  limits  a/,  a/',  is 

if"  dtu  J  &dp; 
the  integral  j  C3dp  being  taken  for  all  positions  of  C  between 


Application  to  Evaluation  of  Definite  Integrals.        373 

two  tangents  to  the  boundary  parallel  to  PQ.  The  question 
is  thus  reduced  to  the  evaluation  of  this  integral ;  which, 
of  course,  is  generally  difficult  enough :  we  may,  however, 
deduce  from  it  a  remarkable  result ;  for  if  the  integral 

be  extended  to  all  possible  positions  of  C,  it  gives  the  whole 
number  of  pairs  of  positions  of  the  points  A,  B  which  lie 
inside  the  area  ;  but  this  number  is  Q,z ;  hence 

JJ(73^^  =  3Qa, 

the  integration  extending  to  all  possible  positions  of  the 
chord  C;  its  length  being  a  given  function  of  its  co-ordinates 
p,w. 

Cor.  Hence  if  L,  12,  be  the  perimeter  and  area  of  any 
closed  convex  contour,  the  mean  value  of  the  cube  of  a  chord 

drawn  across  it  at  random  is  — =-. 

Li 

It  follows  that  if  a  line  cross  such  a  contour  at  random, 
the  chance  that  three  other  lines,  also  drawn  at  random,  shall 

meet  the  first  inside  the  contour,  is  24  — . 

Some  other  cases  of  definite  integrals  deduced  from  the 
theory  of  Probability  are  given  in  a  Paper  in  the  Philo- 
sophical Transactions  for  1868,  pp.  1 81-199.  See  also  Pro- 
ceedings London  Math.  Soc,  vol.  viii. 

Several  Examples  on  Mean  Values  and  Probability  are 
annexed ;  some  of  them,  as  also  some  of  the  questions  which 
have  been  explained  in  this  Chapter,  are  taken  from  the 
Papers  on  the  subject  in  the  Educational  Times,  by  the  Editor, 
Mr.  Miller,  as  also  by  Professor  Sylvester,  Mr.  Woolhouse, 
Col.  Clarke,  Messrs.  Watson,  Savage,  and  others.  Some  few 
are  rather  difficult ;  but  want  of  space  has  prevented  our 
giving  the  solutions  in  the  text. 

"We  may  refer  to  Todhunter's  valuable  History  of  Pro- 
bability for  an  account  of  the  more  profound  and  difficult 
questions  treated  by  the  great  writers  on  the  theory  of  Pro- 
bability. 


374  On  Mean  Value  and  Probability. 


Examples. 

1.  A  chord  is  drawn  joining  two  points  taken  at  random  on  a  circle  :  find  the 
mean  area  of  the  lesser  of  the  two  segments  into  which  it  divides  the  circle. 

Ans. . 

4        » 

2.  Find  the  mean  latitude  of  all  places  north  of  the  Equator. 

Am.  320  .704. 

3.  Find  the  mean  square  of  the  velocity  of  a  projectile  in  vacuo,  taken  at  all 
instants  of  its  flight  till  it  regains  the  velocity  of  projection. 

Ans.   V2  cos2  a  +  |F2  sin2  a :  where  V=  initial  velocity,  and  a  =  angle 
of  projection. 

4.  If  x  and  y  are  two  variahles,  each  of  which  may  take  independently  any 
value  between  two  given  limits  (different  for  each),  show  that  the  mean  value 
of  the  product  xy  is  equal  to  the  product  of  the  mean  values  of  x  and  y. 

5.  If  X,  Y  are  points  taken  at  random  in  a  triangle  ABC,  what  is  the 
chance  that  the  quadrilateral  ABXY  is  convex  ? 

Am.  -. 
3 

For,  it  is  easy  to  see  that  of  the  three  quadrilaterals  ABXY,  ACXY,  BCXY, 
one  must  be  convex,  and  two  re-entrant. 

6.  Find  the  mean  area  of  the  quadrilateral  formed  by  four  points  taken  at 
random  on  the  circumference  of  a  circle. 

Am.  ~  (area  of  circle). 
v 

7.  A  class  list  at  an  examination  is  drawn  up  in  alphabetical  order ;  the  num- 
ber of  names  being  n.  If  a  name  be  selected  at  random,  find  the  chance  that  the 
candidate  shall  not  be  more  than  m  places  from  his  place  in  the  order  of  merit. 

2m  +  I       m(m  +  1)       .„  _,      „,,  .    .  »  ,         ,         .    , 

Am. 5 — -.     UN. 13. — This  is  not,  of  course,  the  value  of  the 

n  n* 

chance  after  the  selection  has  been  made  :  tbis  may  easily  be  found.) 

8.  A  traveller  starts  from  a  point  on  a  straight  river  and  travels  a  certain 
distance  in  a  random  direction.  Having  quite  lost  his  way,  he  starts  again  at 
random  the  next  morning,  and  travels  the  same  distance  as  before.  Find  the 
chance  of  his  reaching  the  river  again  in  the  second  day's  journey. 

Ans.  -. 
4 

9.  Two  lengths,  b,  b',  are  laid  down  at  random  in  a  line  a,  greater  than 
either :  find  the  chance  that  they  shall  not  bave  a  common  part  greater  than  e. 

(a-b-b'  +  c)2 


Ans, 


(a-b)(a-b>) 


Examples.  375 

io.  A  person  in  firing  10  shots  at  a  mark  has  hit  5  times,  and  missed  5.  Find 
the  chance  that  in  the  next  10  shots  he  shall  hit  5  times,  and  miss  5. 

Ans,    — '    ^  •  ' .  =  21—.     If  the  first  10  shots  had  not  heen  fired,  so  that 
19.  17.  13      4199 

nothing  was  known  as  to  his  skill,  the  chance  would  he  — :  if  he 

had  heen  found  to  hit  the  mark  half  the  numher  of  times  out  of  a 

63 
large  numher,  the  chance  would  he  — >. 

11.  If  a  line  I  he  divided  at  random  into  4  parts,  the  mean  square  of  one 

of  the  parts  is  —  I2 :  hut  if  the  line  he  divided  at  random  into  2  parts,  and 
10 

each  part  again  divided  into  2  parts,  then  the  mean  square  of  one  of  the  4  parts 

is  - 1*. 
9 

12.  Three  points  are  taken  at  random  in  a  line  I.  Find  the  mean  distance 
of  the  intermediate  point  from  the  middle  of  the  line. 

Ans.  4^J. 
16 

13.  A  certain  city  is  situated  on  a  river.  The  prohahility  that  a  specified 
inhabitant  A  lives  on  the  right  bank  of  the  river  is,  of  course,  ^,  in  the  absence 
of  any  further  information.  But  if  we  have  found  that  an  inhabitant  B  lives  on 
the  right  bank,  find  the  probability  that  A  does  so  also. 

2 
Ans.  -.     (N.B. — It  is  here  assumed  that  every  possible  partition  of  the 

number  of  inhabitants  into  2  parts,  by  the  river,  is  equally  probable 
a  priori.) 

14.  If  A,  B,  C,  D,  are  four  given  points  in  directum,  and  2  points  are  taken 
at^random  in  AD,  and  one  is  taken  in  BC:  find  the  chance  that  it  shall  fall  be- 
tween the  former  two. 


Ans.  -i-_  l±BC*+BC(AB+CD)  +  2AB.CD\ 


15.  If  z  =  x  +  y,  where  x  may  have  any  value  from  o  to  a,  and  y  any  value 
from  o  to  b :  find  the  probability  that  z  is  less  than  an  assigned  value  c  (suppose 
b<a).     ' 

Ans.  (1)  I£c<b,  pi- — r. 

2ao 

(2)  If  a>c>b,     p2=C-^-. 
a 

nTf                                        («  +  b  -  cf 
(3)If*>«,  J*-i ^ 

If  we  denote  the  functions  expressing  the  probability  in  the  three 
cases  by  /i(«,  b,  c),  /2(«,  b,  c),  f*(a,  b,  c),  we  shall  find  the  rela- 
tion 

/i(«,  h  e)  +fz{a,  b,c)  =/2(«,  b,  c)  +f2(b,  a,  c). 


376  On  Mean  Value  and  Probability. 

1 6.  In  the  cubio  equation 

p  and  q  may  have  any  values  between  the  limits  i  I.     Find  the  chance  that  the 
three  roots  are  real. 

45 

17.  Two  observations  are  taken  of  the  same  magnitude,  and  the  mean  of  the 
results  is  taken  as  the  true  value.  If  the  error  of  each  observation  is  assumed  to 
lie  within  the  limits  +  a,  and  all  its  values  to  be  equally  probable,  show  that  it 
is  an  even  chance  that  the  error  in  the  result  lies  between  the  limits  ±  0.293  «• 

18.  A  point  is  taken  at  random  in  each  of  two  given  plane  areas.  Show 
that  the  mean  square  of  the  distance  between  the  two  points  is 

ft*  +  A'2  +  A2  ; 

where  A  is  the  distance  between  the  centres  of  gravity  of  the  areas ;  and  k,  k' 
are  the  radii  of  gyration  of  each  area  round  its  centre  of  gravity. 

19.  The  mean  square  of  the  area  of  the  triangle  formed  by  joining  any  three 
points  taken  in  any  given  plane  area  is  -  h-  k2  ;  where  h,  k  are  the  radii  of  gyra- 
tion of  the  area  round  the  two  principal  axes  of  rotation  in  its  plane. 

If  one  of  the  points  is  fixed  at  the  centre  of  gravity,  the  value  is  %h2k-. 
(Mr.  Woolhoxjse.) 

20.  A  line  is  divided  at  random  into  3  parts.  Find  the  chance — ( 1)  that  they 
will  form  a  triangle ;  (2)  an  acute-angled  triangle. 

Am.  (1).  pi  =  \. 

^  (2)-  Pi  =  3  log  2  -  2. 

2 1.  A  line  is  divided  into  n  parts.     Find  the  chance  that  they  cannot  form  a 
V_    v^olygon.  ,( 

_°1      =.     An,.     2, 

22.  If  two  stars  are  taken  at  random  in  the  northern  hemisphere,  find  the 
chance  that  their  distance  exceeds  900. 

Am.  -. 

IT 

23.  The  vertices  of  a  spherical  triangle  are  points  taken  at  random  on  a 
sphere.    Find  the  chance — ( 1)  that  all  its  angles  are  acute ;  (2)  that  all  are  obtuse. 

24.  Show  that  the  mean  value  of  -,  where  p  is  the  distance  of  two  point* 

.  & 

taken  at  random  within  a  circle,  is . 

3«r 


Examples.  377 

25.  Two  equal  lines  of  length  a  include  an  angle  0  :  find  the  chance  that  if 
two  points  P,  Q  are  taken  at  random,  one  on  each  line,  their  distance  PQ,  shall  be 
less  than  a. 

Ans.  (1).  When  -  >  B  >  -;  p\  =       .     -  +  2  cos  9. 

v  '  2  3    *       2  sin  0 

,  IT  7T  —   0 

(2).  When  0  >  -  ;        p*  =  — r— . 
v  '  2  2  sin  0 

Here  the  functions  are  connected  by  the  relation  F(9)  +  F(ir  -  0)  =/(0)  +/(tt-0)  . 

26.  The  density  of  a  city  population  varies  inversely  as  the  distance  from  a 
central  point.  Find  the  chance  that  two  inhabitants  chosen  at  random  within  a 
radius  r  from  the  centre  shall  not  live  further  than  a  distance  r  from  each  other. 

ii,  2  /       V3\        1    f2  Odd       3  [2dde 

Ans.  p  = log  3  +  -  ( 1  -  —  1  +  —       -r-t  +  —      -r— ■  ; 

^      3      4     &°      it  \         2/        27rJoSin0      2TrJ'rsin0 

3 

whence  p  =  0.7771.    This  result  is  easily  obtained  by  employing 
the  values  given  in  Question  25. 

27.  Four  points  are  taken  at  random  within  a  circle  or  an  ellipse.  Show 
that  the  chance  that  they  form  a  re-entrant  quadrilateral  is -. 

28.  Find  the  mean  distance  of  two  points  within  a  sphere.       Ans.  — r. 

29.  Three  points  A,  2?,  C  are  taken  within  a  circle,  whose  centre  is  0.  Find 
the  chance  that  the  quadrilateral  ABGO  is  re-entrant. 

1        4 
4      3t2 

30.  Find  the  chance  that  the  distance  of  two  points  within  a  square  shall  not 
exceed  a  side  of  the  square. 

Ans.  p  =  it  — ^-. 
o 

31.  In  the  same  case,  find  the  chance  that  the  distance  shall  not  exceed  an 
assigned  value  c ;  a  being  the  side  of  the  square. 

cz  I  8  1    \ 

Ans.  (1).  When  c<a;  p  —  —  [ira2 ac  +  ~c2] . 

,  v    txt,  <?2    .    ,«         e2      ±2c*+az  ,- — l       c2      c4       I 

(2).  Whenc>a;iJ  =  4-;Sin-1 — ir  -  + r — Vc2-a2-2  — +  -. 

^a2         c         a2     3      a3  a2    zut     3 

32.  Three  points  are  taken  at  random  on  a  sphere;  the  chance  that  in 
the  spherical  triangle  some  one  angle  shall  exceed  the  sum  of  the  other  two 

is  -.     Also  the  chance  that  its  area  shall  exceed  that  of  a  great  circle  is  7. 
2  o 

33.  If  a  line  be  divided  at  random  into  4  parts,  show  that  it  is  an  even  chance 
that  one  of  the  parts  is  greater  than  half  the  line. 

[25] 


378  On  Mean  Value  and  Probability. 

^4.  Prove  that  the  mean  distance  of  a  point  within  a  triangle  from  the  vertex 


I  (a  +  b      (a-*)(a»-J2)      hT- 


+ 


\-         a  +  b  +  e) 
J^aTb^cY 


3(2  2C3 

where  h  is  the  altitude  of  the  triangle.     (See  Ex.  6,  p.  347.) 

35.  The  mean  value  of  the  distance  between  any  two  points  in  an  equilateral 
triangle  is 


Hil+>ei) 


This  question  may  be  solved  by  proving  that  M  =  -  M0,  where  M0  is  the 

mean  distance  of  an  angle  of  the  triangle  from  any  point  within  it.  For,  let 
jlfo  =  M^>  where  /x  is  constant,  and  A  =  area  of  the  triangle.  Take  now  any 
element  dS  of  the  triangle ;  draw  from  it  parallels  to  the  sides  to  meet  the  base ; 
let  5  be  the  area  of  the  equilateral  triangle  so  f ormed :  the  sum  of  the  whole 
number  of  cases  will  be  equal  to 


cffs./itf 


.  dS  =  JfA2, 


if  dS  is  made  to  range  over  the  whole  triangle  :  if  we  call  the  whole  triangle 
unity,  and  put  dS  =  2dad&  as  in  Ex.  3,  p.  344,  5  =  a2,  and  the  integral  be- 
comes —  fi  =  M.     The  result  then  follows  from  34. 

36.  From  a  tower  of  height  h  particles  are  projected  in  all  directions  in 
space  with  a  velocity  due  to  a  falLthrough  h.     Show  that  the  mean  value  of 

the  range  is  M  =  2h  J   ,yl  _  #4  #  ^x. 

(Prof.  "Wolstenholme.) 

37.  In  n  quantities  a,  b,  c,  d  .  .  .  . ,  each  of  which  takes  independently  a 
given  series  of  values  01,  02,  a3,  .  .  .  .  ;  b\,  £2,  ^3,  •  •  .  &c.  (the  number  of  values 
is  different  for  each),  if  we  put 

2a  =  a  +  b  +  c  +  d+...  .&c, 
and  for  shortness  denote  "  the  mean  value  of  x  "  by  Mx,  prove  that 
M 2a  =  Ma  +  M b  +  Mc  +  .  .  .  .  &c.  =  SMa, 
M&af  =  (ZMaY  -  2(Ma)*+  ^M{ai). 

38.  Two  points  are  taken  at  random  in  a  triangle.  Find  the  mean  area  of 
the  triangular  portion  which  the  line  joining  them  cuts  off  from  the  whole 

Ans.  -  of  the  whole. 


Examples.  379 

39.  A  ship  at  A  observes  another  at  B,  whose  course  is  unknown.  Sup- 
posing their  speed  the  same,  prove  that  the  chance  of  their  coming  within  a 

2    .      d 
given  distance  d  of  each  other  is  always  -  sin-1  -,  whatever  the  course  taken 

.""  a  d 

by  A  ;   provided  its  inclination  to  AB  is  not  greater  than  cos"1  -  :    where 

AB  =  a.     (Camb.  Math.  Tripos,  187 1.     Prof.  Miller.) 

40.  A  random  straight  line  crosses  a  circle.  Find  the  chance  that  two 
points  taken  at  random  in  the  circle  shall  lie  on  opposite  sides  of  the  line. 

Ans.  -.    This  is  deduced  at  once  from  the  value  of  M.  the  mean  dis- 

45  t2 

2M 
tance  of  the  two  points  ;  as  the  chance  =  — .     If  two  random  lines 

2irr 

are  drawn,  the  chance  that  both  lines  shall  pass  between  the  points 
.    1 

41.  A  point  0  is  taken  at  random  in  a  triangle.  "What  is  the  probability 
that  if  three  other  points  are  taken  at  random,  one  shall  lie  in  each  of  the  tri- 
angles AOB,  BOC,  CO  A  ? 

Ans.    — .     This  may  easily  be  found  to  depend  on  the  integral  jjafiy .  2da  d&, 
where  a,  j8,  7  are  the  three  triangles  above. 

42.  A  line  crosses  a  circle  at  random  ;  find  the  chance  that  a  point  taken 
at  random  in  the  circle  shall  be  distant  from  the  line  by  less  than  the  radius  of 
the  circle.  s  2 

Ans.   1 . 

3^ 

43.  Two  points  are  taken  on  the  circumference  of  a. semicircle.  Find  the 
chance  that  their  ordinates  fall  on  either  side  of  a  point  taken  at  random  on  the 
diameter.  4 

Ans.  — . 

44.  In  any  convex  area  which  has  a  centre  0,  let  an  indefinite  straight  line 
revolve  round  0,  and  the  locus  of  the  centre  of  gravity  of  either  half  into  which 
it  divides  the  area  be  traced.     Show  that  the  mean  distance  of  0  from  all  points 

in  the  area  is  equal  to  -  the  perimeter  of  this  locus.     Also,  -  of  the  area  enclosed 

4  t  4 

by  this  locus  =  mean  area  of  the  triangle  OXY;  where  X,  Fare  points  taken  at 
random  in  the  given  area.     (Crofton,  Proceedings,  Lond.  Math.  Soc,  vol.  viii.) 

45.  The  probability  that  the  distance  of  two  points  taken  at  random  in  a 
given  convex  area  Cl  shall  exceed  a  given  limit  (a)  is 


-M* 


3a2C+2as)dpdu, 


where  C  is  a  chord  of  the  area,  whose  co-ordinates  are  p,  a ;  the  integration 
extending  to  all  values  of  p,  u>,  which  give  a  chord  C>  a. 

[25  a] 


380  Miscellaneous  Examples. 


Miscellaneous  Examples. 

i.  If  a  be  the  sagitta  of  a  circular  segment  whose  base  is  0,  prove  that  the 
area  of  the  segment  is,  approximately, 

2  .       8  «3 
=  -  ab  +  —  -. 

3  15  * 

2.  Find  the  area  of  the  inverse  of  a  hyperbola,  the  centre  being  the  pole  of 
inversion ;  and  show  that  the  area  of  the  inverse  of  an  ellipse,  under  the  same 
ciroumstances,  is  an  arithmetic  mean  between  the  areas  of  the  circles  described 
on  its  axes  as  diameters. 


^.   ,  A,     •  A        ,    n  dx    la2-  x2 

3.  Find  the  integral  of  —  ,J  — — — . 

x   y  x*  —  0* 


Ans.  tan-1 


la2  -x*      a         ,  b     la2  -  x2 
V^T2  +  *  a^**~b2' 


4.  Prove  that 

f;W  =  ,{-«)/<»iog(;^-:), 

where  |  lies  between  X  and  #0. 

5.  In  a  spiral  of  Archimedes,  if  P,  Q,  and  P*t  Q'be  the  points  of  section  with 
any  two  branches  of  the  curve  made  by  a  line  passing  through  its  pole ;  prove 
that  the  area  bounded  by  the  right  line  and  by  the  two  branches  is  half  the  area 
of  the  ellipse  whose  semiaxes  are  PP'  and  P'  Q. 


6.  Find  the  value  of 


Jdx       \x  +  a 
x  +  c\x  +  b' 


7.  If  an  ellipse  roll  upon  a  right  line,  show  that  the  differential  equation  of 
the  locus  of  its  focus  is 

(y2  +  b*)jx  =  v/ (2«y  +  y2  +  b2)  (zay  -y2-  b2). 

8.  A  circle  rolls  from  one  end  to  the  other  of  a  curved  line  equal  in  length 
to  the  circumference  of  the  circle,  and  then  rolls  back  again  on  the  other  side  of 
the  curve :  prove  that,  if  the  curvature  of  the  curve  be  throughout  less  than 
that  of  the  circle,  the  area  contained  within  the  closed  curve  traced  out  by  the 
point  of  the  circle  which  was  first  in  contact  with  the  fixed  curve  is  six  times 
the  area  of  the  circle.     (Camb.  Math.  Tripos,  1871.) 

9.  In  the  same  case  show  that  the  entire  length  of  the  path  described  is 
eight  times  the  diameter  of  the  circle. 

10.  Prove  that  the  area  of  the  locus  formed  by  the  points  of  intersection  of 
normals  to  an  ellipse,  which  cut  at  right  angles,  is  tt  (a  -  b)*. 


Miscellaneous  Examples.  381 

ii.  Prove  that  the  area  between  two  focal  radii  of  a  parabola  and  the  curve 
is  half  the  area  between  the  curve,  the  corresponding  perpendiculars  on  the 
directrix,  and  the  directrix. 

12.  Evaluate  the  following  integrals  : 

J  V  tana;'     J  J  (i  +  x2)  (i  +  #*) 

13.  If  E  =  (a?  +  axf  +  bx,  and  u  =  log  — ,  find  the  relation 

x2  +  ax  -  VR 

,     .  ,  f  dx         f  x  dx 

between  the  integrals  J  — ,         — 


Vi2       J  Vn 

Am 


cxdx  _  a  cdx       u 


14.  If  a  curve  be  such  that  the  area  between  any  portion  and  a  fixed  right 
line  is  proportional  to  the  corresponding  length  of  the  curve,  show  that  it  is  a 
catenary. 

15.  Prove  that  the  volume  of  a  rectangular  parallelepiped  is  to  that  of  its 
circumscribed  ellipsoid  as  2  :  W3. 

16.  Prove  that        1    —  =  I  — ,  where  sin  ^  =  «  sin  a. 

Jo  vi  - k2 sin20       J0 Vk2 - sin20 

17.  If  any  number  of  triangles  be  inscribed  in  one  ellipse,  and  circumscribed 
to  another  ellipse,  concentric  and  similar,  prove  that  these  triangles  have  all  the 


rb        dy 

18.  Show  that  the  value  of  the  integral  I    - 

J«Vyw»-i 

may  be  exhibited  by  the  following  geometrical  construction.    Let  the  curve 

m            m 
whose  equation  is  rm+2  cos  «  =  1  roll  on  the  axis  of  x ;  take  the  points 

(#1,  y\)  fa,  y-i)  on  the  roulette  described  by  the  pole,  such  that  yi  =  a,  y%  =  b; 
then 

I    ^=====:  =  x%  —  x\.  (Mr.  Jellett.) 

J  a  \/ym  _  j 

19.  If  a  be  the  length  of  the  arc  of  a  spherical  curve  measured  to  any  point 
P,  and  t  be  the  intercept  on  the  great  circle  touching  at  P,  between  the  point  of 
contact  and  the  foot  of  the  perpendicular  from  the  pole,  prove  that 

$  —  t=jsin.pdw. 
The  proof  is  similar  to  that  of  the  corresponding  theorem  in  piano.  See  Art.  158. 


JJ82  Misccl/aneous  Examples. 

20.  Prove  that  the  volume  of  a  polyhedron,  having  for  bases  any  two 
polygons  situated  in  parallel  planes,  and  for  lateral  faces  trapeziums,  is  ex- 
pressed by  the  formula 

where  H  is  the  distance  between  the  parallel  planes,  B  and  B"  the  areas  of  the 
polygonal  bases,  and  B"  the  area  of  the  section  equidistant  from  the  two  bases. 

11.  If  S  be  the  length  of  a  loop  of  the  curve  rn  =  a"cosw0,  and  A  the  are* 
of  a  loop  of  the  curve  r2»  =  a2n  cos  md,  prove  that 

AxS  = . 

2n 

22.  Find  approximately  the  area,  and  also  the  length,  of  a  loop  of  the 
curve  r*  =  J  cos—  .    (See  Biff.  Calc,  Art.  268.) 

Ans.  area  =  a2  x  0.56616;  length  =a  x  2.72638. 

23.  Show  from  Art.  134  that  if  a  parabola  roll  on  a  right  line,  the  locus  of 
its  focus  is  a  catenary. 

24.  If  A  be  the  area  of  any  oval,  B  that  of  its  pedal  with  respect  to  any 
internal  origin  0,  and  C  that  of  the  locus  of  the  point  on  the  perpendicular 
whose  distance  from  0  is  equal  to  the  distance  of  the  point  of  contact  from  0  ; 
prove  that  A,  B,  G  are  in  arithmetical  progression. 

25.  The  arc  of  a  curve  is  connected  with  the  abscissa  by  the  equation  *2  =  kx ; 
find  the  curve. 

a6.  If  the  co-ordinates  of  a  point  on  a  curve  be  given  by  the  equations 

x  =  csiu  20  (1  +  cos  20),       y  =  ecos20(i  -cos  26), 

prove  that  the  length  of  its  arc,  measured  from  its  origin,  is  -  e  sin  30. 

27.  Show  how~to|find  the  sum  of  every  element  of  the  periphery  of  an  ellipse 
divided  by  any  odd  power  (2r+  1)  of  the  semi-diameter  conjugate  to  that  which 
passes  through  the  element,  and  give  the  result  in  the  case  of  the  fifth  power. — 

(Mr.  W.  Roberts.) 


-*»'•  T-A    ,  (2  («2  cos2  0  +  i2  sin2  0)'-1  d9. 
(a*)2r-1Jo 


„..      .        7r(«2  +  02)    . 
This  gives  -i_— 'when  r=  2. 
a3*3 

28.  A  sphere  intersects  a  right  cylinder  ;  prove  that  the  entire  surface  of  the 
cylinder  included  within  the  sphere  is  equal  to  the  product  of  the  diameter  of 
the  cylinder  into  the  perimeter  of  an  ellipse,  whose  axes  are  equal  to  the  greatest 
and  least  intercepts  made  by  the  sphere  on  the  edges  of  the  cylinder. 


Miscellaneous  Examples.  383 

29.  Show  that  the  equations  of  the  involute  of  a  circle  are  of  the  form 

x  =  a  cos  <f>  +  a<f>  sin  <p,       y  =  a  sin  <p  —  af  cos  <p, 

and  prove  that  the  length  of  the  arc  of  this  involute,  measured  from  <f>  =  o,  is 
one  half  of  the  arc  of  a  circle  which  would  be  described  by  a  radius  equal  to  the 
arc  of  its  evolute  moving  through  the  angle  <p. 

30.  Show  that  the  area  of  the  cassinoid 

r*  —  2a2r2  cos  20  +  #4  =  3* 

is  expressed  by  aid  of  an  elliptic  arc,  when  b  >  a  ;  and  by  a  hyperbolic  arc, 
when  a  >  b. 

31.  A  string  AB,  of  given  length,  lies  in  contact  with  a  plane  convex  curve 
with  its  end  A  fixed ;  the  string  is  unwound,  and  B  is  made  to  move  about  A 
till  the  string  is  again  wound  on  the  curve,  the  final  position  of  B  being  B' ; 
prove  that  for  variations  of  the  position  of  A,  the  arc  traced  out  by  B  will  be  a 
maximum  or  a  minimum,  when  the  tangents  at  B  and  B'  are  equally  inclined 
to  the  tangent  at  A  ;  and  will  be  the  former  or  the  latter,  according  as  the  curv- 
ature at  A  is  greater  or  less  than  half  the  sum  of  the  curvatures  at  B  and  B'. — 
(Camb.  Math.  Tripos,  1871.) 

32.  Find  the  value  of      f  -$■  fl'**.  Am.  J-  i  2  ^  . 

Jo  V x  A//3 

33.  Find  the  length,  and  also  the  area,  of  the  pedal  of  a  cissoid,  the  vertex 
being  origin. 

Sa  r  ira* 

Ans.  y  -log(a  + V3)-4»:  2    • 

34.  Prove  that  the  length  of  an  arc  of  the  lemniscate  r2  =  a1  cos  zd  is  repre- 
sented by  the  integral 


a   r  d(p 

fz)  V 1  -  i  sin2  d> 


V2  J  V  I  -  f  8111s  <p 

35.  Integrate  the  equation 

cos  6  (cos  d  -  sin  o  sin  <f>)  dd  +  cos  0  (cos  <p  -  sin  a  sin  d)  d<p  =  o. 

If  thearbitrary  constant  be  determined  by  the  condition  that  the  equation  must 
be  satisfied  by  the  values  (o,  o)  of  (d,  <p),  show  that  the  equation  is  satisfied  by 
putting  0  +  $  =  a. 

36.  Each  element  of  the  surface  of  an  ellipsoid  is  divided  by  the  area  of  the 
parallel  central  section  of  the  surface;  find  the  sum  of  all  the  elementary  quotients 
extended  through  the  entire  ellipsoid.  Ans.  4. 


37.  Hence,  show  that 


n: 


(fi2  -  j/2)  djxdv 


vV-A2  V  F-  M2  VA2-  v3  V  A* 


384 


Miscellaneous  Examples. 


This  depends  on  the  expression  for  an  element  of  the  surface  of  an  ellipsoid  in 
terms  of  the  elliptic  co-ordinates  of  a  point.  See  Salmon's  Geometry  of  Three 
Dimensions,  Art.  41 1.    This  proof  is  due  to  Chasles  (Liouville,  tome  iii.  p.  10). 

38.  Hence,  prove  the  relation 

F(m)  F(n)  +  F(n)  E{m)  -  F(n)  F{m)  =  - 

where 

"■n  n 

F(m)  =  f 2  — —        E{m)  =  (  2  V I  -  w2  sin2  9  dd, 

Jo  V  1  -m2sin20'  Jo 

and  m2  +  w2  ■  1 . 

Let  v=  A sind,  and  fi  =   VA2 'sin2  <f>  +  k2  cos2  <p,   in  the  preceding,  and  it 
becomes 

-    - 
f2  f2       A2  sin2  <p  +  k2  cos2  <p  -  A2  sin2  fl 

Jo  Jo  V/A2sin2^>  +  A2cos2</)  V^2-  A2  sin20 


1  f  V  A2  sin2  ft +  &2  cos2  ft 
V*2-A3sin20 


'rfft  + 


7T       7T 

•2   p2 
0  Jo 


V&2-A2sin2fl 
VA2  sin2  ft  +  A2  cos2  ft 


rfflt/ft 


Jo  Jo  VA2  sin2  ft  + 


A2  dd  d<p 


£2cos2ft  vT2-A2sin20 


This  furnishes  the  required  result  on  making  A  =  mk. 

The  preceding  formula,  which  is  due  to  Legendre,  gives  a  general  relation 
between  complete  elliptic  functions  of  the  first  and  second  species,  with  com- 
plementary moduli.     (Compare  Ex.  7,  8,  p.  331.) 

39.  If  three  curves  be  described  on  the  surface  of  an  ellipsoid,  along  the  first 
of  which  the  perpendicular  to  the  tangent  plane  makes  the  constant  angle  7  with 
the  axis  of  2,  along  the  second  &  with  the  axis  of  y,  and  along  the  third  a  with  the 

axis  of  x,  and  if  the  angles  be  connected  by  the  relations  =  — —  =  ; 

a  0  c 

then,  if  A3,  A2,  A\,  be  the  included  portions  of  the  ellipsoid  surface,  prove  that 

A-i-Az      A1-A2      A-i-Ax 

— ^—  +  — ^— u+       g2       =  °-         (Mr-  Jellett.) 


40.  Show  that  the   results   given  in  Arts.    161  and  162    hold   good  for 
spherical  conies,  where  the  tangents  are  arcs  of  great  circles  on  the  sphere. 


41.  Prove  that 

f«  dx  _  f«   dx 

]»  {(a -*)(*-*)(«-*)}»"  J-  {(a-  *)  (6 -*)(•-*)}" 
where  a,  b,  e  are  in  the  order  of  magnitude. 


Miscellaneous  Examples.  385 

42.  If  a  be  an  imaginary  cube  root  of  unity,  show  that,  if 

_  (»-  a>2)x  +  a)2xs  dp _  (q>  -  aF)  dx 

V  ~  I-«2(a>-a,2)*2'  (I  -  f*)»  (I  +  *>'/)*  ~  (1  -  *2)*  (I  +  <*x2)1' 

(Professor  Cayley.) 

43.  Prove  that  the  value  of 

J™  cos  bx  sin  ax  ,    .  7r  7r 
dx  is  o,    -,    or  -, 
0            x                        4  2 

according  as  J  is  >,  =,  or  <  a. 

44.  Prove  that  : dx  =  -  multiplied  by  the  lesser  of  the 

Jo  z2  2 

numbers  a  and  b. 

45.  If  e  be  the  eccentricity  of  an  ellipse  whose  semiaxis  major  is  unity,  and 
E  the  length  of  its  quadrant,  prove  that 

(- ■  = v  (W.  Roberts.) 

0  (1  -  «2)  V  h2  ~  <?      rn/i-h? 

46.  If  S  represent  the  length  of  a  quadrant  of  the  curve  rm  =  am  cos  md,  and 
Si  the  quadrant  of  its  first  pedal,  prove  that 

2m 
Here  (Ex.  3,  Art.  156),  we  have 

/-  r  (-) 


s=- 


2m         Im  +  1  \ 
I   2»    / 

Also,  since  the  first  pedal  {Biff.  Calc.  Art.  268)  is  derived  by  substituting  ^-^ 
instead  of  m, 

(m+i)fl\/V      \    2m    ) 


81 


2m 


V       am/ 
■+  i)7ra2    F  \2li1)  (m  +  i] 


_  (w+  i)7ra2    x    \2w/       =  (m+  l)ira8 
\        2m} 


r 


386  Miscellaneous  Examples. 

47.  In  general,  if  Sn  be  the  quadrant  of  the  nth  pedal  of  the  curve  in  the  last 
prove  that 

mn+  1 
2m 

Here  it  is  readily  seen  that  the  nth  pedal  is  got  by  substituting  in- 

mn  -f  1 
stead  of  m  in  the  equation  of  the  proposed;  .-.  &c.    (W.  Roberts,  Liouville, 
1845,  P-  I77-) 

48.  If  an  endless  string,  longer  than  the  circumference  of  an  ellipse,  be  passed 
round  the  ellipse  and  kept  stretched  by  a  moving  pencil ;  prove  that  the  pencil 
will  trace  out  a  confocal  ellipse. 

49.  If  two  confocal  ellipses  be  such  that  a  polygon  can  be  inscribed  in  one 
and  circumscribed  to  the  other,  prove  that  an  indefinite  number  of  such  polygons 
can  be  described,  and  that  they  all  have  the  same  perimeter.  (Chasles,  Comp. 
Rend.  1843.) 

50.  To  two  arcs  of  a  hyperbola  whose  difference  is  rectifiable  correspond 
qual  arcs  of  the  lemniscate  which  is  the  pedal  of  the  hyperbola.     {Ibid.) 

51.  Prove  that  the  tangents  drawn  at  the  extremities  of  two  arcs  of  a  conic, 
whose  difference  is  rectifiable,  form  a  quadrilateral  whose  sides  all  touch  the 
same  circle.     {Ibid.) 

52.  In  the  curve 

x%  +  y*  =  a*, 

prove  that  any  tangent  divides  that  portion  of  the  curve  between  two  cusps  into 
two  arcs  which  are  to  each  other  as  the  segments  of  the  portion  of  the  tangent 
intercepted  by  the  axes. 

53.  If  two  tangents  to  a  cycloid  cut  at  a  constant  angle,  prove  that  their 
sum  bears  a  constant  ratio  to  the  arc  of  the  curve  between  them. 

54.  If  AB,  ab,  be  quadrants  of  two  concentric  circles,  their  radii  coincid- 
ing ;  show  that  if  an  arc  Ab  of  an  involute  of  a  circle  be  drawn  to  touch  the 
circles  at  A,  b,  the  arc  Ab  is  an  arithmetical  mean  between  the  arcs  AB  and  ab. 

55.  If  ds  represent  an  infinitely  small  superficial  element  of  area  at  a  point 
outside  any  closed  plane  curve,  and  t,  If  the  lengths  of  the  tangents  from  the 
point  to  the  curve,  and  6  the  angle  of  intersection  of  these  tangents  :  prove  that 

the  sum  of  the  elements  represented  by  — ,  taken  for  all  points  exterior  to 

the  curve,  is  27r2.     (Prof.  Crojton,  Phil.  Trans. ,  1868.) 

56.  Show  that,  for  all  systems  of  rectangular  axes  drawn  through  a  given 
point  in  a  given  plane  area, 

I  \\  ^-y*)dxdy  f  +  4  {  \\*ydxdy  J\ 

taken  over  the  whole  of  the  area,  is  constant ;  and  that  for  a  triangle,  the  point 
being  its  centre  of  gravity,  this  constant  value  is 

(tVa)2  (a4  +  b*  +  ei  -  b*c*  -  c^aT-  -  a*b*). 

Mr.  J.  J.  Walker.) 


Miscellaneous  Examples.  387 

57.  If  ab  —  a'b',  prove  that 

Jo  Jo  xv 

=  log  (?\  log  (^  {^(oo)-^(o)}, 

provided  the  limits  <\>  (o)  and  <p  (00  )  are  both  definite. 

(Mr.  Elliott,  Proceedings,  L6nd.  Math.  Soc,  1876.) 

58.  If  8  denote  the  surface,  and  V  the  volume,  of  the  cone  standing  on  the 
focal  ellipse  of  an  ellipsoid,  and  having  its  vertex  at  an  umbilic  ;  prove  that 

tf  =  ™(33-c2)*,      V=$*c(b2-c2), 

where  a,  b,  e  are  the  principal  semiaxes  of  the  ellipsoid. 

59.  Prove  that,  if  p  be  positive  and  less  than  unity, 


and 


f1  dx  it  1 

(xp  +  x-p)  log  ( 1  +  *)  —  =  — - -a  (1), 

V  *      pampir     p2 

J  {xp  +  x-p)  log  (1  -  x)  --  =  -  cotjjir  -  -  ,  (2), 

Jo  x      p  p6 


where  (1)  may  be  deduced  from  (2)  by  putting  x2  for  x. 

(Prof.  "Wolstenholme.) 

60.  If  /a,  v  be  the  elliptic  co-ordinates  of  a  point  in  a  plane,  prove  that  the 
area  of  any  portion  of  the  plane  is  represented  by 


li 


(fjr  —  v2)  dfidv 


taken  between  proper  limits. 

61.  Prove  that  the  differential  equation,  in  elliptic  co-ordinates,  of  any  tan- 
gent to  the  ellipse  /*  =  /*i  is 

d/x  dv 


VV  -  #)  G"2  -  rf)      V(c2  -  v2)  {fix2  -  v2) 

how  that  the  preceding  differential  equal 
ntegral. 

63.  Prove  that  the  differential  equation  of  the  involute  of  the  ellipse  p  =  fu  is 


62.  Hence  show  that  the  preceding  differential  equation  in  ft  and  v  admits 
of  an  algebraic  integral. 


JSS**J£i?*-* 


388  Miscellaneous  Examples. 

64.  Show  that,  for  a  homogeneous  solid  parallelepiped  of  any  form  and 
dimensions,  the  three  principal  axes  at  the  centre  of  gravity  coincide  in  direction 
with  those  of  the  solid  inscribed  ellipsoid  which  touches  at  the  six  centres  of 
gravity  of  its  six  faces ;  and  that,  for  each  of  the  three  coincident  axes,  and 
therefore  for  every  axis  passing  through  their  common  centre  of  gravity,  the 
moment  of  inertia  of  the  parallelepiped  is  to  that  of  the  ellipsoid  in  the  same 
constant  ratio,  viz.,  that  of  10  to  it. — (Prof.  Townsend.) 

65.  Show  that  the  volumes  of  any  tetrahedron,  and  of  the  inscribed  ellipsoid 
which  touches  at  the  centres  of  gravity  of  its  four  faces,  have  the  same  principal 
axes  at  their  common  centre  of  gravity ;  and  that  their  moments  of  inertia  for  all 
planes  through  that  point  have  the  same  constant  ratio  (viz.  18  V3  :  tt). — (Ibid.) 

66.  A  quantity  M  of  matter  is  distributed  over  the  surface  of  a  sphere  of 
radius  a,  so  that  the  surface  density  varies  inversely  as  the  cube  of  the  distance 
from  a  given  internal  point  S,  distant  b  from  the  centre  ;  prove  that  the  sum  of 
the  principal  moments  of  inertia  of  M at  S  is  equal  to  1M (a2- b2). 

(Camb.  Math.  Tripos,  1876.) 

67.  If     (1  -iaz  +  a*)-*=i  +aXi  +  a2X2  •  •  •  +  anXn  +  •  .  .  ,    prove  that 


f+  XnXmdx  =  o,  f  X 


}dx  =  ——. 

2»+  1 


68.  A  closed  central  curve  revolves  round  an  arbitrary  external  axis  in  its 
plane.  Prove  that  the  moments  of  inertia  I  and  J,  with  respect  to  the  axis  of 
revolution  and  to  the  perpendicular  plane  passing  through  the  centre  of  inertia, 
of  the  solid  generated  by  the  revolving  area,  are  given  respectively  by  the 
expressions 


1=  m  (a2  +  3A2),  /=  m  Ik*  -  -\  • 


where  m  represents  the  mass  of  the  solid,  a  the  distance  of  the  centre  of  the 
generating  area  from  the  axis  of  revolution,  h  and  k  the  radii  of  gyration  of  the 
area  with  respect  to  the  parallel  and  perpendicular  axes  through  its  centre,  and 
I  the  arm  length  of  its  product  of  inertia  with  respect  to  the  same  axes. 

(Prof.  Townsend,  Quarterly  Journal  of  Mathematics,  1879.) 

69.  If   U=  ["{x-z)"-lf{z)dz,  find  the  value  of  ^.  Ans.  f(z). 

Jo  dxn 

70.  Prove  that  the  superficial  area  of  an  ellipsoid  is  represented  by 

2irc2  +  2vab  v  — : . 

J  0  V(l  -  *2*2)  (I  -  tf'2*2) 

where  a2  -  #*  =  aV,     J2  -  c2  =  e72  P. 

(Mr.  Jellett,  Hermathena,  1883.) 

71.  Find  the  mean  distance  of  two  points  on  opposite  sides  of  a  square  whose 
side  is  unity. 


Am.  2.~y_  -flog^+Va). 


Miscellaneous  Examples.  389 

72.  A  cube  being  cut  at  random  by  a  plane,  what  is  the  chance  that  the 
section  is  a  hexagon  ? — (Col.  Clarke.) 

VI  cot-1  V\  -  VI  cot-1  VI  ,  , 

Ans.  — ~ =  .04646. 

73.  Three  points  are  taken  at  random,  one  on  each  of  three  faces  of  a  tetra- 
hedron :  what  is  the  chance  that  the  plane  passing  through  them  cuts  the  f  ourth 
face  ? — (Col.  Clarke.) 

Ans.   -. 
4 

74.  Two  stars  are  taken  at  random  from  a  catalogue :  what  is  the  chance 
that  one  or  both  shall  always  be  visible  to  an  observer  in  a  given  latitude,  A  ? 
— (Ibid.) 

Ans.   -  versin  A  +  -  sin  A. 
2  4 

75.  Find  the  chance  that  the  centre  of  gravity  of  a  triangle  lies  inside  the 
triangle  formed  by  three  points  taken  at  random  within  the  triangle. 


Am'  27(2+ylog4)' 


76.  Two  points  are  taken  at  random  in  a  triangle,  the  line  joining  them 
dividing  the  triangle  into  two  portions :  find  the  mean  value  of  that  portion 
which  contains  the  centre  of  gravity. 

Ans.  —  f  470  -\ log  4  J  =  .6967,  the  triangle  being  unity. 

The  mean  value  of  the  greater  of  the  two  portions  is   — (-  -  log  2  =  .6987. 

77.  Show  that  the  mean  distance  M  of  a  point  in  a  rectangle  from  one  angle 
is  given  by 

lM  =  d  +  —  log  -1—  +  —  log , 

ia  b  2b     °     a 

a  and  b  being  the  sides,  d  the  diagonal. 

78.  Show  that  the  mean  distance  M  of  two  points  within  a  rectangle 
given  by 

__     as      33  1       a2      b2\       5   /£2        a  +  d     a2,      b+d\ 

This  result  may  be  deduced  from  the  preceding ;  for  if  fi  =  mean  distance  of  a 
point  within  the  rectangle  whose  sides  are  x,  y,  from  one  of  its  angles,  it  is  easy 
to  see  that 

a2b2M=^\    I    xyfxdxdy;  .-.  &c. 


390  Miscellaneous  Examples. 

79.  Show  that  if  M  be  the  mean  distance  of  two  points  within  any  convex 
area  XI,  we  have 


=  h\\™ 


dpda>, 


where  2,  2'  are  the  segments  into  which  the  area  is  divided  by  a  straight  line 
crossing  it;  the  co-ordinates  of  the  line  being  p,  w ;  and  the  integration  ex- 
tending to  all  positions  of  the  line. 

This  may  be  seen  by  considering  that  if  a  random  line  crosses  the  area,  the 

chance  of  its  passing  between  the  two  points  is  -— ,  where  L  is  the  length  of  the 

boundary.    Again,  for  any  position  of  the  line,  the  chance  of  the  points  lying 

on  opposite  sides  of  it  is  — -  ;  therefore  the  whole  chance  is  —  M  (22'),  where 

2fi2  fl2  " 

if  (22')  is  the  mean  value  of  the  product  22'  for  all  positions  of  the  line. 


80.  In  the  same  case  we  also  have 
M 


Ssfj**** 


C  being  the  length  of  the  intercepted  chord.     Hence  we  have  the  remarkable 
identity 

(Crofton,  Proceedings,  Lond.  Math.  Soc,  vol.  8.) 

81.  Show  that  if  p  be  the  distance  of  two  points  taken  at  random  in  the 
same  area, 


'(^■sii0"**- 


This  may  be  applied  to  the  circle.    (See  Ex.  24,  p.  376.) 

82.    Show  that  the  mean  area  of  the  triangle  determined  by  three  points 
chosen  at  random  within  any  convex  area  is 


M 


=  n-^3jjtf322«W«, 


where  2  =  either  segment  cut  off  by  the  chord  C ;  but  throughout  the  integra- 
tions, as  the  direction  of  the  chord  alters,  2  means  always  the  segment  on  the 
same  side  of  the  chord  as  at  first. 


INDEX. 


Allman,  on  properties  of  paraboloid, 

268,  281. 
Amsler's  planimeter,  214. 
Annular  solids,  261. 
Approximate,     methods     of    finding 

areas,  211. 
Archimedes,  on  solids,  254. 
spiral  of,  194,  380. 
area  of,  194. 
rectification  of,  227. 
Areas  of  plane  curves,  176. 

Ball,  on  Amsler's  planimeter,  215. 
Bernoulli's  series,  by  integration  by 

parts,  128. 
Binet,  on  principal  axes,  312. 
Buff  on' s  problem,  352. 

Cardioid,  area  of,  192. 

rectification  of,  227,  238. 
Cartesian  oval,  rectification  of,  239. 
Catenary,  equation  to,  183. 
rectification  of,  223. 
surface  of  revolution  by,  260. 
Cauchy,  on  exceptional  cases  in  defi- 
nite integrals,  128. 
on  principal  and  general  values  of 

a  definite  integral,  132. 
on  singular  definite  integrals,  134. 
on  hyperbolic  paraboloid,  271. 
Chasles,  on  rectification  of  ellipse,  234, 
248,  386. 
on  Legendre's  formula,  384. 
Cone,  right,  256. 

Crofton,  on  mean  value  and  probabi- 
lity, 333-379,  387,  390. 
Cycloid,  189. 

Definite  integrals,  30,  115. 
exceptional  cases,  128. 
infinite  limits,  131,  135. 


Definite  integrals,  principal  and  gene- 
ral values,  132. 
singular,  134. 
differentiation  of,  143,  147. 
deduced  by  differentiation,  144. 
integration  under  the  sign  J",  148. 
double,  149,  313. 

Descartes,  rectification  of  oval  of,  239. 

Differentiation  under  the  sign  of  inte- 
gration, 107. 

Dirichlet's  theorem,  316. 

Elliott,  extension  of  Holditch's  theo- 
rem, 209. 

on  Frullani's  theorem,  157,  387. 
Ellipse,  arc  of,  226. 
Ellipsoid,  266. 

quadrature  of,  282. 

of  gyration,  309,  312. 

momental,  309. 

central,  310. 
Elliptic,  integrals,  29,  173,  226,  232, 
235,  243,  279,  384,  387. 

co-ordinates,  249,  387. 
Epitrochoid,  rectification  of,  237. 
Equimomental  cone,  310. 
Errors  of  observation,  361. 
Euler,  102. 

theorem  on  parabolic  sector,  198. 
Eulerian  integrals,  117,  124,  159. 

definition  of — 

T{n)  and£(m,  n),  124,  160. 

T{m  +  n)  * 

T(n)T{i-n)  =  -.-—,  162. 
smmr 

,a.ueofr(I)r(?)...r(^) 

164. 
table  of  log  (m),  169. 


392 


Index. 


Fagnani's  theorem,  229. 
Folium  of  Descartes,  192,  218. 
Frequency,  curve  of,  356. 
Frullani,  theorem  of,  165,  387. 

Gamma  functions,  124,  159. 

Gauss,  on  integration  over  a  closed 

surface,  287. 
Genocchi,    rectification    of    Cartesian 

oval,  240,  242. 
Graves,  on  rectification  of  ellipse,  234. 
Green's  theorem,  326. 
Groin,  269. 
Gudermann,  183. 
Guldin's  theorems,  262,  263,  288. 
Gyration,  radius  of,  293. 

Helix,  rectification  of,  244. 
Hirst,  on  pedals,  202. 
Holditch,  theorem  of,  206. 
Hyperbola,  rectification  of,  233. 
Landen's  theorems  on,  232. 
Hyperbolic  sines  and  cosines,  182. 
Hypotrochoid,  see  epitrochoid. 

Inertia,  integrals  of,  291. 

moments  of,  291. 

products  of,  291,  306. 

principal  axes  of,  307. 

momental  ellipsoid  of,  309 . 
Integrals,  definitions  of,  1,  114. 

elementary,  2. 

double,  149,  313. 

of  inertia,  291. 

transformation  of  multiple,  320. 
Integration,  different  methods  of,  20. 

by  parts,  20. 

of  =*,68. 

Xn-  I 

by  successive  reduction,  63. 
by  differentiation,  71,  144. 
of  binomial  differentials,  75. 
by  rationalization,  92,  97. 
by  differentiation  undersign/,  109. 
by  infinite  series,  110. 
regarded  as  summation,  30,  114. 
double,  269,  313. 
change  of  order  in,  314. 
over  a  closed  surface,  284. 

Jacobians,  323,  326. 
Jellett,  on  quadrature  of  ellipsoid,  283, 
388. 


Zempe,  theorem  on  moving  area,  210. 

Lagrange's  series,  remainder  in,  158. 
Lambert,  theorem  on  elliptic  area,  196. 
Landen,  theorem  on  hyperbolic  arc,  232. 

on  difference  between  asymptote 
and  arc  of  hyperbola,  233. 
Legendre,  on  Eulerian  integrals,  160. 

formula  on  rectification,  228,  369. 

relation  between  complete  elliptic 
functions,  384. 
Leibnitz,  on  Guldin's  theorems,  264. 
Lemniscate,  area  of,  191. 

rectification  of,  384. 
Leudesdorf,  157,  210,  220. 
Limacon,  area  of,  192. 

rectification  of,  237. 
Limits  of  integration,  33,  115. 

Mean  Value  and  probability,  333. 
Mean  Value,  definition  of,  333. 

for  one  independent  variable,  334. 

two  or  more  independent  variables, 
337. 
Method  of  quadratures,  178. 
MiUer,  345. 
Momental  ellipse,  300. 

of  a  triangle,  304. 
Moments  of  inertia,  291. 

relative  to  parallel  axes,  292. 

uniform  rod,  294. 

parallelepiped,  cylinder,  295. 

cone,  296. 

sphere,  297. 

ellipsoid,  298. 

prism,  302. 

tetrahedron,  304. 

solid  ring,  305. 
M'Cullagh,  on  rectification  of  ellipse 
and  hyperbola,  236. 

Neil,  on  semi-cubical  parabola,  224, 

249. 
Newton,  method  of  finding  areas,  177- 
by  approximation,  213. 
on  tractrix,  219. 

Observation,  errors  of,  361. 

Panton,  on  rectification  of  Cartesian 

oval,  240. 
Paraboloid,  of  revolution,  266. 
elliptic,  265,  268. 


Index. 


393 


Partial  fractions,  42. 
Pedal,  area  of,  199. 

of  ellipse,  190. 

Steiner's  theorem  on  area  of,  201. 

Eaabe,  on,  202. 

Hirst,  on,  202. 

Roberts,  on,  386. 
Planimeter,  Amsler's,  214. 
Popoff,   on  remainder  in  Legrange's 

series,  159. 
Probability,  used  to  find  mean  values, 

343. 
Probabilities,  349. 
Products  of  inertia,  301,  306. 

Quadrature,  plane,  176. 
on  the  sphere,  276. 
of  surfaces,  279. 
paraboloid,  280. 
ellipsoid,  282. 

Eaabe,  theorem  on  pedal  areas,  202. 
Radius  of  gyration,  293. 
Random  straight  lines,  368. 
Rectification  of,  plane  curves,  222. 

parabola,  223. 

catenary,  233. 

semi-cubical  parabola,  224. 

of  evolutes,  224. 

arc  of  ellipse,  226. 

hyperbola,  231. 

epitrochoid,  237. 

roulettes,  238. 

Cartesian  oval,  239,  247. 

twisted  curves,  243. 
Recurring  biquadratic  under  radical 

sign,  101. 
Reduction,  integration  by,  63. 

by  differentiation,  71,  80. 
Roberts,  W.,  on  Cartesian  oval,  240. 

on  pedals,  386. 
Roulette,  quadrature  of,  205. 
rectification  of,  238. 


Simpson's  rules  for  areas,  213. 
Sphere,  surface  and  volume  of,  252. 

quadrature  on,  276. 
Spheroid,  surface  of,  257,  258. 
Spiral,  hyperbolic,  191. 

of  Archimedes,  194,  227,  380. 

logarithmic,  227. 
Steiner,  theorem  on  pedal  areas,  201. 

on  areas  of  roulettes,  203. 

on  rectification  of  roulettes,  238. 
Surface  of,  solids,  250. 

cone,  251. 

sphere,  252. 

revolution,  254. 

spheroid,  prolate,  257. 
oblate,  258. 

annular  solid,  261. 

Taylor's  theorem,  obtained  by  integra- 
tion by  parts,  126. 
remainder  as  a  definite  integral, 
127. 
Townsend,  on  moments  of  inertia  of  a 
ring,  305,  388. 

on  moments  of  inertia  in 
general,  310. 
Tractrix,  area  of,  219. 

length  of,  225. 

Van  Huraet,  on  rectification,  249. 
Viviani,  Florentine  enigma,  278. 
Volumes  of  solids,  250,  264,  286. 

Wallis,  value  for  ir,  122. 

"VVeddle,  on  areas   by  approximation, 

213. 
Woolhouse,    on  Holditch's  theorem, 

206. 


Zolotareff,  on  remainder  in  Lagrange' 
series,  158. 


THE  END. 


V 


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